Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications
aa r X i v : . [ m a t h . O C ] D ec Conic Mixed-Binary Sets: Convex HullCharacterizations and Applications
Fatma Kılın¸c-Karzan
Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA, [email protected]
Simge K¨u¸c¨ukyavuz
Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL 60208, USA,[email protected]
Dabeen Lee
Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea, [email protected]
We consider a general conic mixed-binary set where each homogeneous conic constraint involves an affinefunction of independent continuous variables and an epigraph variable associated with a nonnegative func-tion, f j , of common binary variables. Sets of this form naturally arise as substructures in a number of appli-cations including mean-risk optimization, chance-constrained problems, portfolio optimization, lot-sizingand scheduling, fractional programming, variants of the best subset selection problem, and distributionallyrobust chance-constrained programs. When all of the functions f j ’s are submodular, we give a convex hulldescription of this set that relies on characterizing the epigraphs of f j ’s. Our result unifies and generalizes anexisting result in two important directions. First, it considers multiple general convex cone constraints insteadof a single second-order cone type constraint. Second, it takes arbitrary nonnegative functions instead of aspecific submodular function obtained from the square root of an affine function. We close by demonstratingthe applicability of our results in the context of a number of broad problem classes. Key words : Conic mixed-binary sets, conic quadratic optimization, convex hull, submodularity, extendedpolymatroid inequalities, fractional binary optimization, best subset selection, distributionally robustoptimization
1. Introduction
In this paper we study the following set S ( f, K ) := { ( x, z ) ∈ R m × { , } n : ∃ y ∈ R s.t. y ≥ f ( z ) , Ax + By ∈ K } , (1)where f : { , } n → R + is a nonnegative function, K is a convex cone containing the origin, and A, B are any matrices of appropriate dimension. Throughout, for a ∈ Z + , we let [ a ] := { , . . . , a } .The set S ( f, K ) we consider is a generalization of the two fundamental sets studied byAtamt¨urk and G´omez (2020) defined as follows: H := ( x, z ) ∈ R m + × { , } n : s σ + X i ∈ [ n ] c i z i + X j ∈ [ m − d j x j ≤ x m , (2) ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications R := ( x, z ) ∈ R m + × { , } n : σ + X i ∈ [ n ] c i z i + X j ∈ [ m − d j x j ≤ x m − x m , (3)where c ∈ R n + , d ∈ R m − , and σ ∈ R + . Atamt¨urk and G´omez (2020) demonstrate that the set H naturally arises in mean-risk minimization and chance-constrained programs, and the set R appearsas a substructure in robust conic quadratic interdiction, lot-sizing and scheduling, queueing systemdesign, binary linear fractional problems, Sharpe ratio maximization, and portfolio optimization.The main contribution of Atamt¨urk and G´omez (2020) is deriving the convex hulls of H and R .In Section 2, we prove that the convex hull of the set S ( f, K ) for any nonnegative function f and arbitrary cone K is given by the following: b S ( f, K ) := { ( x, z ) ∈ R m × [0 , n : ∃ y ∈ R + s.t. ( y, z ) ∈ conv(epi( f )) , Ax + By ∈ K } , (4)where given a function f , epi( f ) denotes its epigraph, i.e., epi( f ) := { ( y, z ) ∈ R × { , } n : y ≥ f ( z ) } and conv(epi( f )) denotes its convex hull. Therefore, the convex hull of S ( f, K ) is given by theinequalities describing conv(epi( f )) and the homogeneous conic constraint Ax + By ∈ K . Thischaracterization highlights, in particular, that the challenge in developing the convex hull of S ( f, K )is solely determined by the complexity of the convex hull characterization of the epigraph of thefunction f . In fact, we prove a more general result that covers this characterization as a specialcase. In particular, we consider a set with multiple conic constraints, each of which involves anaffine function of independent continuous variables and an epigraph variable associated with anonnegative function of common binary variables.There are a number of cases where characterizing the convex hull of conv(epi( f )) iseasy. For example, when f is a nonnegative submodular function, imposing the condi-tion that ( y, z ) ∈ conv(epi( f )) is equivalent to applying the extended polymatroid inequal-ities (Atamt¨urk and Narayanan 2008, Lov´asz 1983). We review the extended polymatroidinequalities and their generalization to arbitrary set functions, namely the polar inequalities,(Atamt¨urk and Narayanan 2020) in Section 3. In Section 4, we discuss how our results can beapplied to more general functions and non-homogeneous conic constraints.The set S ( f, K ) naturally arises in a number of other applications including fractional binaryoptimization, best subset selection, and distributionally robust optimization. We will discuss thesein Section 5. Furthermore, the sets H and R studied by Atamt¨urk and G´omez (2020) are indeedspecial cases of S ( f, K ) where K is taken to be the direct product of a second order cone (SOC)and the nonnegative orthant, and the function f ( z ) is restricted to be the square root of an affinefunction of z . We elaborate this connection in Section 5.1. Consequently, even in the case of a singleconic constraint and a single function f , our convex hull characterization immediately generalizes ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications the results from Atamt¨urk and G´omez (2020) in two directions: (1) by allowing general convexcones K as opposed to the standard SOC, and (2) by allowing a general nonnegative function f ( z )as opposed to the specific one studied by Atamt¨urk and G´omez (2020). We discuss applications ofour framework with multiple conic constraints and multiple functions in Section 5.2. In particular,this generalization allows us cover the fractional optimization models that appear in a broadrange of applications including modeling multinomial logit (MNL) choice models in assortmentoptimization, set covering, market share based facility location, stochastic service systems, bi-clustering, and optimization of boolean query for databases. We discuss another application of ourgeneralization in best subset selection in machine learning in Section 5.3. In this application, thefunction f is an exponential function, which demonstrates the applicability of our result beyond thesquare root function. Furthermore, our results in this context pave the way to use standard solversfor this problem all the while exploiting the submodular structure as opposed to the approachof G´omez and Prokopyev (2020) based on parameterizing the fractional objective function andapplying a customized Newton-type method. We also consider an application in distributionallyrobust chance-constrained programming under Wasserstein ambiguity in Section 5.4 and show thatour results can be used to strengthen the mixed-integer conic reformulation for the case of mixed-binary decision variables, thereby generalizing an existing strengthening that assumes pure binarydecisions in the original chance constraint.
2. Convex hull characterization
We next examine a generalization of our set defined by S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) := (cid:26) ( x, z ) ∈ R mp × { , } n : ∃ y ∈ R p s.t. y j ≥ f j ( z ) , ∀ j ∈ [ p ] ,A j x j + B j y j ∈ K j , ∀ j ∈ [ p ] (cid:27) , (5)where the functions f j : { , } n → R + for j ∈ [ p ] are nonnegative functions and K j for j ∈ [ p ] arecones. A j and B j for j ∈ [ p ] are matrices of appropriate dimensions. The lengths of the continuousvectors x , . . . , x p may be different, but we focus on the setting of equal lengths for simplicity. Ourtheoretical developments can be easily extended to the case of heterogeneous lengths.Note that the following is a convex relaxation of S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ): b S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) := (cid:26) ( x, z ) ∈ R mp × [0 , n : ∃ y ∈ R p s.t. ( y, z ) ∈ conv (cid:0) G (cid:0) { f j } j ∈ [ p ] (cid:1)(cid:1) ,A j x j + B j y j ∈ K j , ∀ j ∈ [ p ] (cid:27) , (6)where G (cid:0) { f j } j ∈ [ p ] (cid:1) := { ( y, z ) ∈ R p × { , } n : y j ≥ f j ( z ) , ∀ j ∈ [ p ] } . (7)We note that the functions f j for j ∈ [ p ] take the same binary variables z . The constraint y j ≥ f j ( z ) gives rise to the epigraph of f j for each j , and therefore, G (cid:0) { f j } j ∈ [ p ] (cid:1) can be viewed as the ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications “intersection” of the epigraphs. Our main result establishes that, under minor assumptions, theconvex hull of the set S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) is indeed given precisely by b S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ).To prove our convex hull result, we first establish a technical proposition that applies to moregeneral mixed-integer sets. We consider sets of the form Q ( G , { K j } j ∈ [ p ] ) := (cid:8) ( x, y, z ) ∈ R mp × R p × { , } n : ( y, z ) ∈ G , A j x j + B j y j ∈ K j , ∀ j ∈ [ p ] (cid:9) , (8)where G ⊆ R p + × { , } n and K j ’s for j ∈ [ p ] are cones. We further assume that G| z =¯ z := G ∩ { ( y, z ) : z = ¯ z } for any fixed ¯ z ∈ { , } n is convex, which means that G| z =¯ z is the face of conv( G ) definedby z = ¯ z . Note that G (cid:0) { f j } j ∈ [ p ] (cid:1) is indeed contained in R p + × { , } n , as f j for j ∈ [ p ] are non-negative functions. Moreover, G (cid:0) { f j } j ∈ [ p ] (cid:1) | z =¯ z for a fixed ¯ z ∈ { , } n is defined by linear con-straints only, so it is convex. Observe that S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) is precisely the projection of Q ( G (cid:0) { f j } j ∈ [ p ] (cid:1) , { K j } j ∈ [ p ] ) onto the ( x, z )-space. Given this relation, to derive our promised convexhull result for the set S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) in the original space, we first establish the followingproposition for sets of the form Q ( G , { K j } j ∈ [ p ] ) defined in the extended space. Proposition 1.
Consider Q ( G , { K j } j ∈ [ p ] ) defined as in (8) for some G ⊆ R p + × { , } n such that G| z =¯ z for any fixed ¯ z ∈ { , } n is convex. Suppose K j is a convex cone for each j ∈ [ p ] . Then, conv( Q ( G , { K j } j ∈ [ p ] )) = b Q ( G , { K j } j ∈ [ p ] ) , where b Q ( G , { K j } j ∈ [ p ] ) := (cid:8) ( x, y, z ) ∈ R mp × R p × [0 , n : ( y, z ) ∈ conv( G ) , A j x j + B j y j ∈ K j , ∀ j ∈ [ p ] (cid:9) . Proof of Proposition 1.
For brevity, define Q := Q ( G , { K j } j ∈ [ p ] ) and b Q := b Q ( G , { K j } j ∈ [ p ] ).The containment conv( Q ) ⊆ b Q is immediate because Q ⊆ b Q and b Q is convex. To prove the reversecontainment, we show that the recessive directions of b Q are also recessive directions of conv( Q )and that the extreme points of b Q are contained in Q .Let ( d x , d y , d z ) be a recessive direction of b Q . Then, A j d jx + B j ( d y ) j ∈ K j for j ∈ [ p ] and ( d y , d z ) isa recessive direction of conv( G ). Suppose for a contradiction that ( d x , d y , d z ) is not a recessive direc-tion of conv( Q ). Then there exists (¯ x, ¯ y, ¯ z ) ∈ conv( Q ) such that (¯ x, ¯ y, ¯ z ) + ( d x , d y , d z ) conv( Q ). As(¯ x, ¯ y, ¯ z ) ∈ conv( Q ), it can be written as a convex combination of some points, denoted ( x i , y i , z i ) for i ∈ I , in Q . Hence, (¯ x, ¯ y, ¯ z ) = P i ∈ I α i ( x i , y i , z i ) for some α ≥ P i ∈ I α i = 1. Since ( x i , y i , z i ) ∈Q , we have A j ( x i ) j + B j ( y i ) j ∈ K j for j ∈ [ k ], and therefore, A j ( x i + d x ) j + B j ( y i + d y ) j ∈ K j for j ∈ [ k ]. Since ( d y , d z ) is a recessive direction of conv( G ), ( y i + d y , z i + d z ) ∈ conv( G ). Moreover,conv( G ) ⊆ R p + × [0 , n implies that d z = 0, so ( y i + d y , z i + d z ) ∈ conv( G ) ∩ { ( y, z ) : z = z i } = G| z = z i ⊆G . Therefore, ( x i + d x , y i + d y , z i + d z ) ∈ Q for i ∈ I , which implies that (¯ x, ¯ y, ¯ z ) + ( d x , d y , d z ) = ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications P i ∈ I α i ( x i + d x , y i + d y , z i + d z ) ∈ conv( Q ), a contradiction. Therefore, ( d x , d y , d z ) is a recessivedirection of conv( Q ).Next, consider an extreme point (ˆ x, ˆ y, ˆ z ) of b Q . Then, (ˆ y, ˆ z ) ∈ conv( G ) and A j ˆ x j + B j ˆ y j ∈ K j holds for all j ∈ [ p ]. Also, as conv( G ) ⊆ R p + × [0 , n , ˆ y ∈ R p + . We claim that (ˆ y, ˆ z ) must be in G . We will prove this by showing that (ˆ y, ˆ z ) must be an extreme point of conv( G ). Assume forcontradiction that there exist distinct points (¯ y, ¯ z ) ∈ conv( G ) and (˜ y, ˜ z ) ∈ conv( G ) such that (ˆ y, ˆ z ) = (¯ y, ¯ z ) + (˜ y, ˜ z ). From conv( G ) ⊆ R p + × [0 , n , we deduce that ¯ y, ˜ y ∈ R p + . Moreover, if for some index j ∈ [ p ] we have ˆ y j = 0, we must also have ¯ y j = ˜ y j = 0. For each j ∈ [ p ], define ¯ x j := ¯ y j ˆ y j ˆ x j wheneverˆ y j >
0, and ¯ x j := ˆ x j whenever ˆ y j = 0. Consider any j ∈ [ p ]. Then, when ˆ y j = 0, we have ¯ y j = 0 aswell as ¯ x j = ˆ x j , and thus (¯ x j , ¯ y j ) = (ˆ x j , ˆ y j ) and hence A j ¯ x j + B j ¯ y j = A j ˆ x j + B j ˆ y j ∈ K j . When ˆ y j > x j , ¯ y j ) = ¯ y j ˆ y j (ˆ x j , ˆ y j ). Moreover, because A j ˆ x j + B j ˆ y j ∈ K j holds, ¯ y j ˆ y j ∈ R + and K j is a cone, we deduce that A j ¯ x j + B j ¯ y j ∈ K j as well. Hence, in either case we conclude that(¯ x, ¯ y, ¯ z ) ∈ b Q . Similarly, for each j ∈ [ p ], define ˜ x j := ˜ y j ˆ y j ˆ x j whenever ˆ y j >
0, and ˜ x j := ˆ x j wheneverˆ y j = 0. As before, we deduce that (˜ x, ˜ y, ˜ z ) ∈ b Q . Finally, for each j ∈ [ p ] such that ˆ y j > x j + ˜ x j ) = 12 (cid:18) ¯ y j ˆ y j ˆ x j + ˜ y j ˆ y j ˆ x j (cid:19) = ¯ y j + ˜ y j y j ˆ x j = ˆ x j , where the last equation follows from (ˆ y, ˆ z ) = (¯ y, ¯ z ) + (˜ y, ˜ z ). Also, for each j ∈ [ p ] such that ˆ y j = 0,we have ¯ x j = ˜ x j = ˆ x j . Thus, (ˆ x, ˆ y, ˆ z ) = (¯ x, ¯ y, ¯ z ) + (˜ x, ˜ y, ˜ z ). This contradicts the fact that (ˆ x, ˆ y, ˆ z )is an extreme point of b Q . Therefore, (ˆ y, ˆ z ) is an extreme point of conv( G ), and thus we must have(ˆ y, ˆ z ) ∈ G . Hence, we have shown that every extreme point of b Q is contained in Q , as required. (cid:3) Proposition 1 is instrumental in proving the following main theorem that gives the convex hullcharacterization of S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ). Theorem 1.
For each j ∈ [ p ] , let f j : { , } n → R + be a nonnegative function and K j be a convexcone containing the origin. Then, the convex hull of S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) defined in (5) is describedby b S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) as defined in (6) . Proof of Theorem 1.
First, we observe that G := G (cid:0) { f j } j ∈ [ p ] (cid:1) ⊆ R p + × { , } n as f j ’s are non-negative functions and that S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) = Proj x,z (cid:0) Q ( G , { K j } j ∈ [ p ] ) (cid:1) . Moreover, G| z =¯ z forany z = ¯ z is convex. Then, Proposition 1 implies thatconv (cid:0) Q ( G , { K j } j ∈ [ p ] ) (cid:1) = (cid:26) ( x, y, z ) ∈ R mp × R p × [0 , n : ( y, z ) ∈ conv( G ) ,A j x j + B j y j ∈ K j , ∀ j ∈ [ p ] (cid:27) . Finally, recall that the convex hull and projection operations commute, i.e., the projection of theconvex hull of a set is equal to the convex hull of the projection of a set. Therefore,conv (cid:0) S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) (cid:1) = conv (cid:0) Proj x,z (cid:0) Q ( G , { K j } j ∈ [ p ] ) (cid:1)(cid:1) ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications = Proj x,z (cid:0) conv (cid:0) Q ( G , { K j } j ∈ [ p ] ) (cid:1)(cid:1) = (cid:26) ( x, z ) ∈ R mp × [0 , n : ∃ y ∈ R p s.t. ( y, z ) ∈ conv( G ) ,A j x j + B j y j ∈ K j , ∀ j ∈ [ p ] (cid:27) = b S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) . (cid:3) The interesting applications of Theorem 1 arise whenever it is relatively easy to give the explicitconvex hull description of the intersections of the epigraphs of nonnegative functions f j : { , } n → R + . One such case is when f j ’s are nonnegative and submodular, where conv(epi( f j ))’s are describedby the extended polymatroid inequalities . When f j ’s are general set functions, a recent workof Atamt¨urk and Narayanan (2020) discusses the polar inequalities that generalize the extendedpolymatroid inequalities. Hence, the polar inequalities are valid for S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) for anyset of nonnegative functions f j ’s. We will briefly review these inequalities in the next section.Next, we focus on the p = 1 case, introduced in Section 1, where we consider a single set function f and a cone K . To simplify our notation in this case, we define S ( f, K ) := S ( { f } , { K } ) = { ( x, z ) ∈ R m × { , } n : ∃ y ∈ R s.t. y ≥ f ( z ) , Ax + By ∈ K } . Consider the set S ( f, K ) := { ( x, z ) ∈ R m × { , } n : Ax + Bf ( z ) ∈ K } . (9)Then, clearly S ( f, K ) ⊆ S ( f, K ), and we arrive at the following immediate corollary of Theorem 1. Corollary 1.
Let f : { , } n → R + be a nonnegative function and K be a convex cone containingthe origin. Then, conv( S ( f, K )) ⊆ b S ( f, K ) . If S ( f, K ) = S ( f, K ) further holds, then conv( S ( f, K )) = b S ( f, K ) . Proof of Corollary 1.
By Theorem 1 applied to the p = 1 case, we deduce conv( S ( f, K )) = b S ( f, K ). Since S ( f, K ) ⊆ S ( f, K ), it follows that conv( S ( f, K )) ⊆ b S ( f, K ). If we have S ( f, K ) = S ( f, K ), then conv( S ( f, K )) = b S ( f, K ) indeed holds, as required. (cid:3) In general, the condition S ( f, K ) = S ( f, K ) does not always hold. To illustrate, take a point(¯ x, ¯ z ) ∈ S ( f, K ). Since S ( f, K ) ⊆ S ( f, K ), we know that (¯ x, ¯ z ) ∈ S ( f, K ). Moreover, by definition of S ( f, K ), we deduce that there exists ¯ y ∈ R such that ¯ y ≥ f (¯ z ) and A ¯ x + B ¯ y ∈ K . Then, because f isa nonnegative function, we have ¯ y ≥
0. Moreover, using the fact that K is a cone we conclude that( α ¯ x, ¯ z ) ∈ S ( f, K ) for any α ≥
1. However, A ¯ x + Bf (¯ z ) ∈ K does not guarantee that A ( α ¯ x ) + Bf (¯ z ) ∈ K if α >
1, which means that ( α ¯ x, ¯ z ) is not necessarily contained in S ( f, K ). This in turn indicatesthat conv( S ( f, K )) is not equal to b S ( f, K ) in general.Nevertheless, there are some important examples, wherein the condition S ( f, K ) = S ( f, K ) indeedholds and thus conv( S ( f, K )) = b S ( f, K ). The following proposition provides a necessary and suffi-cient condition for conv( S ( f, K )) = b S ( f, K ). ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications Proposition 2. S ( f, K ) = S ( f, K ) if and only if the following condition is satisfied:every ( x, z ) satisfying Ax + Bf ( z ) ∈ K also satisfies A ( αx ) + Bf ( z ) ∈ K for any α ≥ . ( ⋆ ) Proof of Proposition 2. ( ⇒ ) Take ( x, z ) such that Ax + Bf ( z ) ∈ K . By definition, we have( x, z ) ∈ S ( f, K ). Then, by S ( f, K ) = S ( f, K ), we have ( x, z ) ∈ S ( f, K ). Now, since ( αx, z ) ∈ S ( f, K )for any α ≥
1, once again using S ( f, K ) = S ( f, K ), we deduce ( αx, z ) ∈ S ( f, K ). Hence, A ( αx ) + Bf ( z ) ∈ K for any α ≥ ( ⇐ ) It suffices to prove that S ( f, K ) ⊆ S ( f, K ). To this end, take ( x, z ) ∈ S ( f, K ). Then thereexists y ≥ y ≥ f ( z ) and Ax + By ∈ K . If y = 0, then f ( z ) = 0 as f is a nonnegativefunction. This implies that Ax + f ( z ) = Ax + By ∈ K , so ( x, z ) ∈ S ( f, K ). If y >
0, it followsfrom A ( f ( z ) /y ) x + Bf ( z ) = ( f ( z ) /y )( Ax + By ) ∈ K . In this case, since y/f ( z ) ≥
1, the premise ofthis direction implies that Ax + Bf ( z ) = A ( y/f ( z ))( f ( z ) /y ) x + Bf ( z ) ∈ K , and therefore, ( x, z ) ∈S ( f, K ). Thus, S ( f, K ) ⊆ S ( f, K ), as required. (cid:3) We highlight a useful sufficient condition that implies Condition ( ⋆ ). Remark 1.
Let K be a convex cone containing the origin. Suppose K , A and B are such thatfor any ( x, z ) satisfying Ax + Bf ( z ) ∈ K , we also have Ax ∈ K .For any ( x, z ) with Ax + Bf ( z ) ∈ K and any α ≥
1, we have ( α − Ax ∈ K , so A ( αx ) + Bf ( z ) =( α − Ax + ( Ax + Bf ( z )) ∈ K . Therefore, Condition ( ⋆ ) is immediately satisfied.We discuss in Section 5 some applications where Condition ( ⋆ ) holds. In particular, usingRemark 1, we will show in Section 5.1 that H and R defined in (2) and (3) satisfy Condition ( ⋆ )and consider other applications in Sections 5.2 – 5.4.Corollary 1 and Proposition 2 can be extended to the case of multiple functions. Consider S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) := (cid:8) ( x, z ) ∈ R mp × { , } n : A j x j + B j f j ( z ) , ∀ j ∈ [ p ] (cid:9) , (10)which is a subset of S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) defined in (5). Corollary 2.
Let f j : { , } n → R + be a nonnegative function and K j be a convex conecontaining the origin for j ∈ [ p ] . Then, conv( S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] )) ⊆ b S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) . If S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) = S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) further holds, then conv( S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] )) = b S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) . Proof of Corollary 2.
Similar to the proof of Corollary 1. (cid:3)
Moreover, we can characterize when S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) = S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ). ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications Proposition 3. S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) = S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) if and only if the following holds:every ( x, z ) satisfying A j x j + B j f j ( z ) ∈ K j for all j ∈ [ p ] also satisfies A j α j x j + B j f j ( z ) ∈ K j for any α j ≥ for all j ∈ [ p ] . ( ⋆⋆ ) Proof of Proposition 3.
Similar to the proof of Proposition 2. (cid:3)
In Section 5.2, inspired by fractional binary programming, we study a generalization of H and R that takes multiple conic quadratic constraints, whose convex hull can be characterized basedon Corollary 2.We close this section by noting that the structure of the set S ( f, K ) allows us to easily embedconstraints on the continuous variables x as well. Remark 2 (Additional constraints on continuous variables).
Let C be a convex conecontaining the origin, and consider the set along with its transformation given by n ( x, z ) ∈ R m × { , } n : ∃ y ∈ R s.t. y ≥ f ( z ) , ˜ Ax + ˜ By ∈ ˜ K , Cx ∈ C o = { ( x, z ) ∈ R m × { , } n : ∃ y ∈ R s.t. y ≥ f ( z ) , Ax + By ∈ K } , where we set K = ˜ K × C , A = [ ˜ A ; C ], and B = [ ˜ B ; 0]. Note that through this representation, wededuce that the additional conic constraints on the continuous variables x can easily be embeddedinto our desirable form of the set S ( f, K ).
3. The extended polymatroid inequalities and the polar inequalities
In this section, we discuss two classes of inequalities, namely, the extended polymatroid inequalitiesand the polar inequalities, which can be used to describe conv(epi( f )) completely or partially (andthus these give either conv( S ( f, K )) or remain valid for it) when f has desirable structure.Given a set function f : 2 [ n ] → R , where 2 [ n ] is the power set of [ n ], let the associated polyhedron of f be defined as P f := { π ∈ R n : π ( V ) ≤ f ( V ) , ∀ V ⊆ [ n ] } , (11)where π ( V ) := P i ∈ V π i and π ( ∅ ) = 0. By slight abuse of notation, throughout, we refer to a setfunction f : 2 [ n ] → R also as f : { , } n → R , where f ( V ) := f ( V ) for V ⊆ [ n ] and V denotes thecharacteristic vector of V . When f is a submodular function, i.e., f satisfies f ( S ) + f ( S ) ≥ f ( S ∪ S ) + f ( S ∩ S ) , ∀ S , S ⊆ [ n ] , P f is called the extended polymatroid of f . Note that P f is nonempty if and only if f ( ∅ ) ≥ f need not satisfy f ( ∅ ) ≥
0. Nevertheless, we can take f − f ( ∅ ) instead so that ( f − f ( ∅ ))( ∅ ) = 0. Hence, P f − f ( ∅ ) is always nonempty. Hereinafter, we use notation ˜ f to denote f − f ( ∅ )for any set function f . In particular, ˜ f is submodular when f is submodular. ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications The associated polyhedron P f is instrumental in generating valid inequalities for epi( f ) due to aclose polarity relation between P f and conv(epi( f )). (We refer the reader to Nemhauser and Wolsey(1988, Chapter I.4.5) for a review of how the concept of polarity is used to obtain facets of apolyhedron.) From this relation, Atamt¨urk and Narayanan (2020) show that the so-called polarinequalities y − f ( ∅ ) ≥ π ⊤ z, ∀ π ∈ P ˜ f (12)are valid for ( y, z ) ∈ conv(epi( f )). Atamt¨urk and Narayanan (2020) also prove that inequalities (12)are facet-defining for conv(epi( f )) if and only if π is an extreme point of P ˜ f . Therefore, the extremepoints of P ˜ f characterize facet-defining polar inequalities. In the case of a general set function f ,the inclusion relationship in (12) is strict (see Example 1, Atamt¨urk and Narayanan 2020), whichmeans that the polar inequalities may not be sufficient to describe the convex hull of epi( f ).When f is submodular, the polar inequalities are precisely the extended polymatroid inequali-ties (Atamt¨urk and Narayanan 2008). Moreover, it is well-known that when f is submodular, theextended polymatroid inequalities indeed provide a complete description of conv(epi( f )). Theorem 2 (Lov´asz (1983), Atamt¨urk and Narayanan (2008, Theorem 1)) . Let f : { , } n → R be a submodular function. Then, conv(epi( f )) = (cid:8) ( y, z ) ∈ R × [0 , n : y − f ( ∅ ) ≥ π ⊤ z, ∀ π ∈ P ˜ f (cid:9) . Edmonds (1970) provides the following explicit characterization of the extreme points of P ˜ f . Theorem 3 (Edmonds (1970)) . Let f : { , } n → R be a submodular function. Then π ∈ R n is an extreme point of P ˜ f if and only if there exists a permutation σ of [ n ] such that π σ ( t ) = f ( V t ) − f ( V t − ) , where V t = { σ (1) , . . . , σ ( t ) } for t ∈ [ n ] and V := ∅ by definition. We note that when f is not submodular, there are extreme points π of P f are not necessarily ofthe form given in Therorem 3. The proof of Theroem 3 yields an O ( n log n ) time algorithm forseparating a violated extended polymatroid inequality given a point (¯ y, ¯ z ) ∈ R × R n , which amountsto solving max π (cid:8) ¯ z ⊤ π : π ∈ P ˜ f (cid:9) (Atamt¨urk and Narayanan 2008, Section 2).These results on submodular functions lead us to the following corollary of Theorem 1. Corollary 3.
Suppose f : { , } n → R + is a nonnegative submodular function and K is a convexcone containing the origin. Then, conv( S ( f, K )) is given by the extended polymatroid inequalitiesfor f − f ( ∅ ) and the homogeneous conic constraint Ax + By ∈ K . Proof of Corollary 3.
This is a direct consequence of Theorems 1 and 2. (cid:3) ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee:
Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications Corollary 4.
For each j ∈ [ p ] , let f j : { , } n → R + be a nonnegative submodular function and K j be a convex cone containing the origin. Then, the convex hull of S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) definedin (5) is described by b S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) as defined in (6) , and moreover b S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) = (cid:26) ( x, z ) ∈ R mp × [0 , n : ∃ y ∈ R p s.t. ( y j , z ) ∈ conv(epi( f j )) ,A j x j + B j y j ∈ K j , ∀ j ∈ [ p ] (cid:27) . Proof of Corollary 4.
Let G := { ( y, z ) ∈ R p × { , } n : y j ≥ f j ( z ) , ∀ j ∈ [ p ] } . Then,
G ⊆ R p + ×{ , } n as f j ’s are nonnegative functions. The result follows from Theorem 1 and the following fact.When f j is a nonnegative submodular function for each j ∈ [ p ], from Theorem 2 of Baumann et al.(2013) (see also, Proposition 1, Kılın¸c-Karzan et al. 2019) we deduce thatconv( G ) = { ( y, z ) ∈ R p × [0 , n : ( y j , z ) ∈ conv(epi( f j )) , ∀ j ∈ [ p ] } . (cid:3) We refer the reader to Edmonds (1970) and Lov´asz (1983) for a list of basic submodular functionsas well as operations preserving submodularity.
Remark 3.
We close this section by emphasizing that our result holds not only for a functionof the form q σ + P i ∈ [ n ] c i z i for z ∈ { , } n , but also for general submodular functions, such as g ( σ + P i ∈ [ n ] c i z i ) where g is concave. In particular, the constant elasticity of substitution func-tion, (cid:16)P i ∈ [ n ] c pi z pi (cid:17) /p for any c ∈ R n + and p ≥
4. Extensions
Our results are applicable in the cases where the conic constraint is non-homogeneous, or thefunction f ( x ) does not satisfy the nonnegativity assumption, or where we have only a partial convexhull description of the epigraph of f ( x ) available. Remark 4 (Non-homogeneous conic constraints).
Our results are still of interest when S ( f, K ) has a non-homogeneous constraint, i.e., Ax + By + C ∈ K for some C = 0 instead of Ax + By ∈ K . Indeed, by adding a new variable and an affine constraint, we can always rewritethis set using a homogeneous conic constraint. That is, { ( x, z ) ∈ R m × { , } n : ∃ y ∈ R s.t. y ≥ f ( x ) , Ax + By + C ∈ K } = { ( x, z ) ∈ R m × { , } n : ∃ y, v ∈ R s.t. y ≥ f ( x ) , Ax + By + Cv ∈ K , v = 1 } . (13)Here, { ( x, v, z ) ∈ R m × R × { , } n : ∃ y ∈ R s.t. y ≥ f ( x ) , Ax + By + Cv ∈ K } is of the form S ( f, K ),and the set in (13) is obtained from the intersection of this set and an affine hyperplane definedby v = 1 after projecting out v . Therefore, (cid:8) ( x, z ) ∈ R m × { , } n : ∃ ( y, v ) ∈ R s.t. ( y, z ) ∈ conv(epi( f )) , Ax + By + Cv ∈ K , v = 1 (cid:9) (14)is a valid convex relaxation for the set in (13) by Theorem 1. ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications While Remark 4 is useful, due to the presence of the affine constraint v = 1 in (14), (14) may notprovide a convex hull description of the set in (13). This is demonstrated in the following example. Example 1.
Note that { ( x, z ) ∈ R × { , } : p x + z + z ≤ x − } = { ( x, z ) ∈ R × { , } : ( √ z + z , x , x − ∈ L } is of the form S ( f, K ) where f ( z ) = √ z + z . In addition, f is submodular, f ( ∅ ) = 0, and P f = n π ∈ R : π ≤ , π ≤ , π + π ≤ √ o , with extreme points ( π , π ) = (1 , √ −
1) and ( π , π ) = ( √ − , ( ( x, z ) ∈ R × { , } : ∃ ( y, v ) ∈ R + × R s.t. y ≥ ( √ − z + z , y ≥ z + ( √ − z , p x + y ≤ x − v, v = 1 ) . (15)In Appendix A, we show that (15) has an extreme point where either z or z is fractional. Remark 5 (Arbitrary set functions).
Consider the set S ( h, K ) where h is neither nonneg-ative nor nonpositive. Let h min := min z ∈{ , } n h ( z ). Then, the function f ( z ) := h ( z ) − h min is anonnegative function and we have S ( h, K ) = { ( x, z ) ∈ R m × { , } n : Ax + Bh ( z ) ∈ K } = { ( x, z ) ∈ R m × { , } n : Ax + Bf ( z ) + Bh min ∈ K } . Now, we can apply the transformations from Remark 4 and use Corollary 1 to arrive at the desiredrepresentation.
Remark 6 (Supermodular functions).
Suppose the function of interest is supermodular.Let h : { , } n → R be a supermodular function and K be a convex cone containing the ori-gin. If h is nonnegative, then one would want to apply Corollary 1 on the set S ( h, K ) = { ( x, z ) ∈ R m × { , } n : Ax + Bh ( z ) ∈ K } and convexify it by introducing an auxiliary variable y and replacing Ax + Bh ( z ) ∈ K with the constraints ( y, z ) ∈ conv(epi( h )) and Ax + By ∈ K . However,we do not know how to describe conv(epi( h )) when h is supermodular. Nevertheless, we can stillutilize our technique of transforming the homogeneous conic constraint into a non-homogeneousone and applying Remark 4. To this end, define h max := max z ∈{ , } n { h ( z ) } and f ( z ) := − h ( z ) + h max .Then, we know that f is a nonnegative submodular function, and we have S ( h, K ) = { ( x, z ) ∈ R m × { , } n : Ax + B ( h max − f ( z )) ∈ K } = { ( x, z ) ∈ R m × { , } n : ∃ v ∈ R s.t. Ax + Bh max v − Bf ( z ) ∈ K , v = 1 } . ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications Therefore, as f ( z ) is submodular, we can add the extended polymatroid inequalities for f tostrengthen the set S ( h, K ).We note that this transformation applied to a supermodular function is indeed easy as computing h max amounts to minimizing a submodular function, which can be done in polynomial time. Remark 7.
As discussed in Section 3, for general (not necessarily submodular) f ,Atamt¨urk and Narayanan (2020) introduced the class of polar inequalities that are valid for b S ( f, K )for any nonnegative function f , which can be used to strengthen the continuous relaxation.
5. Applications
In this section, we present several optimization problems in which sets of the form S ( f, K ) appearas a substructure. In this respect, we highlight two forms of objective functions that immediatelylead to our desired structure:(i) min p f ( z ) + k Dx + d k ,(ii) min k Dx + d k f ( z ) ,where f ( z ) is a nonnegative function, in addition to norm constraints of the form k ( z ; Dx ) k ≤ t .In Section 5.1, we discuss the implications of our results in the context of applications fromAtamt¨urk and G´omez (2020) which have the objectives of form (i) where f is submodular, and inSection 5.2 we explore their use in the case of fractional programming. In Section 5.3, we pointout connections with variants of best subset selection problem (G´omez and Prokopyev 2020) inwhich the problems have objectives of the form (ii) and the function f is supermodular. Finally,in Section 5.4, we highlight the connection of our work with substructures exploited within thecontext of distributionally robust chance constrained problems involving binary variables. Recall the following sets studied by Atamt¨urk and G´omez (2020): H = ( x, z ) ∈ R m + × { , } n : s σ + X i ∈ [ n ] c i z i + X j ∈ [ m − d j x j ≤ x m , R = ( x, z ) ∈ R m + × { , } n : σ + X i ∈ [ n ] c i z i + X j ∈ [ m − d j x j ≤ x m − x m . In this section, we let the function f : { , } n → R + to be defined by f ( z ) := q σ + P i ∈ [ n ] c i z i .Note that f is submodular, provided that σ + P i ∈ [ n ] c i z i is nonnegative for z ∈ { , } n . Hence, byTheorem 2, conv(epi( f )) is described by the extended polymatroid inequalities for f − √ σ and ≤ z ≤ . Motivated by this observation, Atamt¨urk and G´omez (2020) prove the following resultsfor the sets H and R by analyzing KKT conditions of a generic linear optimization problem overthe domain H or R . ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications Proposition 4 (Atamt¨urk and G´omez (2020, Proposition 5)) . Let f ( z ) = q σ + P i ∈ [ n ] c i z i where σ + P i ∈ [ n ] c i z i ≥ for z ∈ { , } n , and let H be defined as in (2) . Then conv( H ) = ( x, z ) ∈ R m + × R n : ∃ y ∈ R + s.t. ( y, z ) ∈ conv(epi( f )) , s y + X i ∈ [ m − d i x i ≤ x m . Proposition 5 (Atamt¨urk and G´omez (2020, Proposition 6)) . Let f ( z ) = q σ + P i ∈ [ n ] c i z i where σ + P i ∈ [ n ] c i z i ≥ for z ∈ { , } n , and let R be defined as in (3) . Then conv( R ) = ( x, z ) ∈ R m + × [0 , n : ∃ y ∈ R + s.t. ( y, z ) ∈ conv(epi( f )) , y + X i ∈ [ m − d i x i ≤ x m − x m . We next show that these results are simple corollaries of Theorem 1. Recall L k +1 = { ( ξ, y ) ∈ R k × R : y ≥ k ξ k } denotes the SOC in R k +1 . Throughout, let e i be the unit vector of appropriatedimension with a one in the i th component, and let Diag( · ) represent a diagonal matrix with thespecified diagonal entries. Corollary 5.
Theorem 1 implies Proposition 4.
Proof of Corollary 5.
Note that the mixed-integer set H considered in Proposition 4 satisfies H = ( x, z ) ∈ R m + × { , } n : s ( f ( z )) + X j ∈ [ m − d j x j ≤ x m = n ( x, z ) ∈ R m + × { , } n : h f ( z ); p d x ; . . . ; p d m − x m − ; x m i ∈ L m +1 o = n ( x, z ) ∈ R m + × { , } n : ˜ Ax + ˜ Bf ( z ) ∈ L m +1 o , where ˜ B := e ∈ R m +1 and ˜ A := [0 ⊤ ; Diag( √ d ; . . . ; p d m − ; 1)] ∈ R ( m +1) × m . Note that if ( x, z )satisfies ˜ Ax + ˜ Bf ( z ) ∈ L m +1 , then due to the structure of ˜ A, ˜ B and the cone L m +1 , we deduce that x satisfies ˜ Ax ∈ L m +1 as well. Then, by Remark 1, Condition ( ⋆ ) holds. Therefore, by Proposition 2, H = n ( x, z ) ∈ R m + × { , } n : ∃ y ∈ R s.t. y ≥ f ( z ) , ˜ Ax + ˜ By ∈ L m +1 o . The result then follows by applying Theorems 1–3, Corollary 4, and Remark 2 so that we canhandle the nonnegativity constraint on the continuous variables x ∈ R m + (which is nothing but avery simple conic constraint) in addition to the SOC constraint ˜ Ax + ˜ By ∈ L m +1 . (cid:3) Corollary 6.
Theorem 1 implies Proposition 5.
Proof of Corollary 6.
Note that y + P i ∈ [ m − d i x i ≤ x m − x m is a rotated SOC constraintgiven by h y ; p d x ; . . . ; p d m − x m − ; x m − − x m ; x m − + x m i ⊤ ∈ L m +2 . ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications By defining ˜ B := e ∈ R m +2 and ˜ A := [0 ⊤ ; Diag( √ d ; . . . ; p d m − ); 0 ⊤ ; 0 ⊤ ] + e m [0 , . . . , , , −
1] + e m +1 [0 , . . . , , , ∈ R ( m +2) × m , this constraint is equivalent to ˜ Ax + ˜ By ∈ L m +2 . Then, R = ( x, z ) ∈ R m + × { , } n : ( f ( z )) + X j ∈ [ m − d j x j ≤ x m − x m = n ( x, z ) ∈ R m + × { , } n : ˜ Ax + ˜ Bf ( z ) ∈ L m +2 o = n ( x, z ) ∈ R m + × { , } n : ∃ y ∈ R s.t. y ≥ f ( z ) , ˜ Ax + ˜ By ∈ L m +2 o , where the last line follows from Proposition 2 (exactly as in the case of Corollary 5). Furthermore,by applying the transformation from Remark 2 so that we can handle the nonnegativity constrainton the continuous variables x ∈ R m + in addition to the SOC constraint ˜ Ax + ˜ By ∈ L m +2 , we concludethat the result follows from Theorem 1, Corollary 4, and Theorems 2 and 3. (cid:3) Recall that Theorem 1 may take multiple functions into account at the same time. Based on Theo-rem 1, we can characterize the convex hull of a set defined by multiple conic quadratic constraints,generalizing the results on H and R . For two finite sets H, R of indices, the following set is definedby | H | conic quadratic constraints of the type used in H and | R | constraints of the type used in R : M := ( x, z ) : ( x, z ) ∈ R m ( | H | + | R | )+ × { , } n , s σ ℓ + X i ∈ [ n ] c ℓ,i z i + X j ∈ [ m − d ℓ,j x ℓ,j ≤ x ℓ,m , ∀ ℓ ∈ Hσ ℓ + X i ∈ [ n ] c ℓ,i z i + X j ∈ [ m − d ℓ,j x ℓ,j ≤ x ℓ,m − x ℓ,m , ∀ ℓ ∈ R . (16)We are interested in conv( M ). As in the proofs of Corollary 5 and 6, we can rewrite M as M := ( ( x, z ) ∈ R m ( | H | + | R | )+ × { , } n : ˜ A ℓ x ℓ + ˜ B ℓ f ℓ ( z ) ∈ L m +1 , ∀ ℓ ∈ H ˜ A ℓ x ℓ + ˜ B ℓ f ℓ ( z ) ∈ L m +1 , ∀ ℓ ∈ R ) , where x ℓ = ( x ℓ, , . . . , x ℓ,m ) ⊤ and ˜ A ℓ , ˜ B ℓ are defined as in the proofs of Corollary 5 and 6 for ℓ ∈ H ∪ R .Notice that M is of the form S ( { f j } j ∈ [ p ] , { K j } j ∈ [ p ] ) defined as in (10). Moreover, as both H and R satisfy Condition ( ⋆ ), M satisfies the condition ( ⋆⋆ ). Therefore, we can apply Corollary 2 andProposition 3 to obtain the following proposition characterizing the convex hull of M . Proposition 6.
For ℓ ∈ H ∪ R , let f ℓ ( z ) = q σ ℓ + P i ∈ [ n ] c ℓ,i z i where σ ℓ + P i ∈ [ n ] c ℓ,i z i ≥ for z ∈{ , } n , and let M be defined as in (16) . Then conv( M ) = ( x, z ) : ( x, z ) ∈ R m ( | H | + | R | )+ × [0 , n , ∃ y ℓ ∈ R + s.t. ( y ℓ , z ) ∈ conv(epi( f ℓ )) , s y ℓ + X j ∈ [ m − d ℓ,j x ℓ,j ≤ x ℓ,m , ∀ ℓ ∈ H ∃ y ℓ ∈ R + s.t. ( y ℓ , z ) ∈ conv(epi( f ℓ )) , y ℓ + X j ∈ [ m − d ℓ,j x ℓ,j ≤ x ℓ,m − x ℓ,m , ∀ ℓ ∈ R . ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications We remark that Proposition 6 generalizes Propositions 4 and 5. For the rest of this section, we listsome applications of Proposition 6.
Remark 8 (fractional binary programs).
We can use M to model optimization problemsof the following form: min z ∈X X ℓ ∈ R a ℓ, + P i ∈ [ n ] a ℓ,i z i b ℓ, + P i ∈ [ n ] b ℓ,i z i , (17)where X ⊆ { , } n and a ℓ, , a ℓ,i , b ℓ, , b ℓ,i for i ∈ [ n ] are all nonnegative numbers. The fractional opti-mization model (17) is used in a wide range of application domains including modeling multinomiallogit (MNL) choice models in assortment optimization, set covering, market share based facilitylocation, stochastic service systems, bi-clustering, and optimization of boolean query for databases(see, Borrero et al. 2017, and references therein). We can reformulate (17) by introducing an aux-iliary variable for each fraction. Note that a ℓ, + P i ∈ [ n ] a ℓ,i z i b ℓ, + P i ∈ [ n ] b ℓ,i z i ≤ u ℓ ⇔ a ℓ, + X i ∈ [ n ] a ℓ,i z i ≤ u ℓ v ℓ , v ℓ = b ℓ, + X i ∈ [ n ] b ℓ,i z i . Then (17) is equivalent tomin z ∈{ , } n , u,v ∈ R ℓ X ℓ ∈ R u ℓ : 4 a ℓ, + X i ∈ [ n ] a ℓ,i z i ≤ u ℓ v ℓ , ∀ ℓ ∈ R,v ℓ = b ℓ, + X i ∈ [ n ] b ℓ,i z i , ∀ ℓ ∈ R . In particular, 4 a ℓ, + P i ∈ [ n ] a ℓ,i z i ≤ u ℓ v ℓ for ℓ ∈ R give rise to a set of the form M . G´omez and Prokopyev (2020) study the following general model for the
Best Subset Selection (BSS)problem: min β ∈ R n ,z ∈{ , } n ( k a − U β k g ( P i ∈ [ n ] z i ) : − M z i ≤ β i ≤ M z i , ∀ i ∈ [ n ] ) , (18)where a ∈ R k , U ∈ R k × n are data, M ∈ R + corresponds to big-M value, and g : R + → R + is anon-increasing convex function. The best subset selection problem in linear regression is to finda sparse subset of regressors that best fits the data. To this end, in Problem (18), the regressionvariables are β and each binary variable z i determines whether a regressor β i is selected. Whilemean squared error (MSE) is a popular criterion to measure the goodness of fit, other criteria suchas Akaike Information Criterion (AIC), corrected AIC (AICc), and Bayesian Information Criterion(BIC) have also been proposed. The latter three criteria have desirable properties that addresssome shortcomings of the MSE criterion, but they involve (non-convex) logarithmic terms in the ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications objective function that call for advanced solution methods. In particular, for AIC and BIC, wehave g ( z ) = e − α P i ∈ [ n ] z i , and for AICc we have g ( z ) = e − α/ ( α − P i ∈ [ n ] z i ) for an appropriate choice of α ≥
0. We refer the reader to G´omez and Prokopyev (2020) for more details.G´omez and Prokopyev (2020) work with the following reformulation of (18):min β ∈ R n ,z ∈{ , } n ,s ∈ R + k a − U β k s : s ≤ g ( X i ∈ [ n ] z i ) , − M z i ≤ β i ≤ M z i , ∀ i ∈ [ n ] . (19)In (19), s ≤ g ( P i ∈ [ n ] z i ) is equivalent to − s ≥ − g ( P i ∈ [ n ] z i ), and since − g ( P i ∈ [ n ] z i ) is submod-ular, the feasible region of (19) can be strengthened by applying the extended polymatroidinequalities for − g . However, the objective of (19) is a still fractional function. Consequently,G´omez and Prokopyev (2020) apply a customized Newton-type method after parameterizing thefraction.In contrast, we observe here that (18) admits a SOC reformulation with a linear objective andwhose feasible region contains a substructure that fits our framework. Define h ( z ) := g ( P i ∈ [ n ] z i ).By introducing a new variable t ∈ R + to capture the objective function, we observe that (18) isequivalent to the following problem:min t ∈ R + ,β ∈ R n ,z ∈{ , } n (cid:8) t : t · h ( z ) ≥ k a − U β k , − M z i ≤ β i ≤ M z i , ∀ i ∈ [ n ] (cid:9) . Note that the nonlinear constraint t · h ( z ) ≥ k a − U β k in this formulation is equivalent to requiring[2 a − U β ; t − h ( z ); t + h ( z )] ∈ L k +2 . Furthermore, when we define the function h ( z ) := g ( P i ∈ [ n ] z i ) based on a nonnegative non-increasing convex function g , we deduce that h ( z ) is nonnegative and supermodular. By usingthe transformation from Remark 6, we define h max := max z ∈{ , } n { h ( z ) } (note that h max ≥
0) and f ( z ) := − h ( z ) + h max , and arrive at the equivalent conic constraint[2 a − U β ; t + f ( z ) − h max ; t − f ( z ) + h max ] ∈ L k +2 , where f is a nonnegative submodular function. In order to homogenize this constraint as we didin Remark 4, we introduce another decision variable v ∈ R and the constraint v = 1. By letting x = [ t ; v ; β ] and m = n + 2, our conic constraint becomes ˜ Ax + ˜ Bf ( z ) ∈ L k +2 , where we select˜ A = a − U − h max h max and ˜ B = k − . Hence, we arrive at the equivalent problemmin x ∈ R m ,z ∈{ , } n n x : ˜ Ax + ˜ Bf ( z ) ∈ L k +2 , x ∈ R + , x = 1 , − M z i ≤ x i +2 ≤ M z i , ∀ i ∈ [ n ] o , ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications where f ( z ) = − g ( P i ∈ [ n ] z i ) + max ¯ z ∈{ , } n { g ( P i ∈ [ n ] ¯ z i ) } is a nonnegative submodular function. Con-sider a point ( x, z ) that satisfies ˜ Ax + ˜ Bf ( z ) ∈ L k +2 . Then, x = [ t ; v ; β ] and t · ( h max v − f ( z )) ≥k a − U β k holds. Note that this constraint along with t ≥ h max v − f ( z ) ≥
0. More-over, since f is a nonnegative function, we have h max v − ≥ h max v − f ( z ). Then, t · ( h max v − ≥ t · ( h max v − f ( z )) ≥ k a − U β k holds, i.e., x satisfies ˜ Ax ∈ L k +2 as well. Then, Remark 1 implies thatCondition ( ⋆ ) holds and we can thus apply Proposition 2 to arrive at the equivalent formulationmin x ∈ R m ,y ∈ R + ,z ∈{ , } n n x : y ≥ f ( z ) , ˜ Ax + ˜ By ∈ L k +2 , x ∈ R + , x = 1 , − M z i ≤ x i +2 ≤ M z i , ∀ i ∈ [ n ] o . Our developments in this application highlight that one can exploit the submodularity structurein this problem all the while using the standard optimization solvers without the need to developspecialized algorithms such as Newton-type methods or parametrization of the fractional objective.
Distributionally robust chance-constrained programming (DR-CCP) under Wasserstein ambiguityis formulated as min ( x,z ) ( c ⊤ [ x ; z ] : sup P ∈F N ( θ ) P [ ξ
6∈ W ( x, z )] ≤ ǫ, ( x, z ) ∈ X ) . (20)Here, c ∈ R n + m is a cost vector, x ∈ R m is a vector of continuous decision variables, z is a vector of n binary decision variables, X ⊂ R n + m is a compact domain for the decision variables, W ( x, z ) ⊆ R K is a decision-dependent safety set, ξ ∈ R K is a vector of K random variables with distribution P ∗ ,and ǫ ∈ (0 ,
1) is the risk tolerance for the random variable ξ falling outside the safety set W ( x, z ).Because the distribution P ∗ is often unavailable, independent and identically distributed (i.i.d.)samples { ξ i } i ∈ [ N ] are drawn from P ∗ to approximate P ∗ using the empirical distribution P N on thesesamples. To address the ambiguity in the true distribution, distributionally robust optimizationmodel (20) considers the worst-case probability of violating the safety constraints over a set ofdistributions on R K , given by F N ( θ ) , that contains the empirical distribution P N where θ is aparameter that governs the size of the ambiguity set, and the degree of conservatism of (20).Chen et al. (2018) and Xie (2019) show that DR-CCP can be formulated as a mixed-integerconic program for certain W ( x, z ), thus enabling their solution with standard optimization solvers.However, these MIP reformulations are difficult to solve in certain cases for which the continuousrelaxations provide weak lower bounds. Ho-Nguyen et al. (2020a,b) consider a mixed-integer linear substructure of the mixed-integer conic formulation of DR-CCP. The authors propose valid linearinequalities and other enhancements that strengthen the continuous relaxation bounds and scaleup the sizes of the problem that can be solved.While previous research focused on the linear constraints, the strengthening of the mixed-integer conic reformulation of DR-CCP considered by Xie (2019) focuses on constraints of the form ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications k [ η x ; η z ; η ] k ∗ ≤ t , where t is a continuous epigraph variable, η , η ∈ { , } are constants with η + η ≥
1, and k · k ∗ is the dual of a norm k · k . To obtain a strengthening from this conic constraint,we consider the case when η = 1, i.e., when there is so-called left-hand side uncertainty. The normis used to measure the Wasserstein distance and it is typical to consider an ℓ q -norm, whose dualnorm is an ℓ p -norm with p + q = 1. Xie (2019) pays specific attention to the pure-binary case, i.e., m = 0 where the continuous variables x are not present inside the norm. In this particular case,Xie (2019) observes that the function k [ η z ; η ] k p is a submodular function and the correspondingextended polymatroid inequalities can be applied to strengthen the set { ( z, t ) : k [ η z ; η ] k p ≤ t } and this approach results in significant computational benefit. When the continuous variables x arepresent, although k [ η x ; η z ; η ] k p is not even a set function. Nevertheless, our framework appliesto constraints of the form k [ η x ; η z ; η ] k p ≤ t . To elaborate, let K m +2 p denote the p -th order conein R m +2 , and define f ( z ) := k [ z ; η ] k ∗ . Note that as z pj = z j for all j ∈ [ n ] and η /p = η as well, andthus from Remark 3 we deduce that f is a nonnegative submodular function. Moreover, (cid:8) ( x, t, z ) ∈ R m + × R + × { , } n : k [ η x ; η z ; η ] k p ≤ t (cid:9) = (cid:8) ( x, t, z ) ∈ R m + × R + × { , } n : [ f ( z ); x ; . . . ; x m ; t ] ∈ K m +2 p (cid:9) = (cid:8) ( x, t, z ) ∈ R m + × R + × { , } n : ∃ y ∈ R s.t. y ≥ f ( z ) , [ f ( z ); x ; . . . ; x m ; t ] ∈ K m +2 p (cid:9) = n ( x, t, z ) ∈ R m + × R + × { , } n : ∃ y ∈ R s.t. y ≥ f ( z ) , ˜ A ( x ; t ) + ˜ By ∈ K m +2 p o , where ˜ B := e ∈ R m +2 and ˜ A := [0 ⊤ ; Diag(1; . . . ; 1)] ∈ R ( m +2) × ( m +1) . Here the second equationfollows from Proposition 2 because Condition ( ⋆ ) holds (for any ( x, z ) satisfying ˜ Ax + ˜ Bf ( z ) ∈ K m +2 p , due to the structure of ˜ A, ˜ B and the cone K m +2 p , we have x satisfies ˜ Ax ∈ K m +2 p as well,implying that we can apply Remark 1 to deduce that Condition ( ⋆ ) holds). Therefore, we can applyCorollary 1. Consequently, our results indicate that it is possible to exploit submodularity in theDR-CCP context even when l we have both continuous and binary decision variables. In particular,if ( x, z ) is a mixed-binary decision vector, then by using Theorem 1, Corollaries 1 and 4, andTheorems 2 and 3 we can strengthen the resulting reformulation of DR-CCP under Wassersteinambiguity. Acknowledgments
This research is supported, in part, by ONR grant N00014-19-1-2321, the Institute for Basic Science (IBS-R029-C1), and NSF grant CMMI 1454548.
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Suppose for a contradiction that every extreme point of (15) has binary z and z components.Consider a point (¯ x, ¯ z ) in (15) with (¯ z , ¯ z , ¯ y ) = (1 / , / , √ /
2) and ¯ x = 1 + p ¯ x + 1 /
2. Because¯ z is not binary, (¯ x, ¯ z ) should not be an extreme point of (15), and thus it must be a convexcombination of extreme points of (15). First, we show that the extreme points that give rise to(¯ x, ¯ z ) must satisfy ( z , z , y ) = (0 , ,
0) or ( z , z , y ) = (1 , , √ I , be the extreme pointsof (15) with ( z , z ) = (0 , x ; z ; y ) = ( x ,i , x ,i , , ,
0) for i ∈ I , . Similarly define I , as the set of extreme points of (15) with ( z , z ) = (1 , x ,i , x ,i , , , √
2) for i ∈ I , ; I , as the set of extreme points of (15) with ( z , z ) = (1 , x ,i , x ,i , , ,
1) for i ∈ I , ,and I , as the set of extreme points of (15), given by ( x ,i , x ,i , , ,
1) for i ∈ I , . We know that(¯ x , ¯ x , ¯ z , ¯ z , ¯ y ) is a convex combination of these extreme points with the associated multipliers α ij,k ≥ , i ∈ I j,k , j, k ∈ { , } and P j,k ∈{ , } P i ∈ I j,k α ij,k = 1. Let α j,k = P i ∈ I j,k α ij,k , for j, k ∈ { , } .Then, since ¯ z = 1 / z = 1 /
2, we must have α , + α , = α , + α , = 1 /
2, and √
22 = α , √ α , + α , . Substituting α , = α , = 1 / − α , and solving the above equation for α , , we obtain α , = 1 / α , and α , = α , = 0, as desired.Moreover, (¯ x , ¯ x , ¯ z , ¯ z , ¯ y ) = X i ∈ I , α i , ( x ,i , x ,i , , ,
0) + X j ∈ I , α j , ( x ,j , x ,j , , , √ , and P i ∈ I , α i , = P j ∈ I , α j , = 1 /
2. Then, ¯ x = P i ∈ I , α i , x ,i + P j ∈ I , α j , x ,j . Note that¯ x = X i ∈ I , α i , x ,i + X j ∈ I , α j , x ,j ≥ X i ∈ I , α i , (1 + | x ,i | ) + X j ∈ I , α j , (1 + q x ,j + 2) ≥ X i ∈ I , α i , | x ,i | + X j ∈ I , α j , q x ,j + 2 . (21) ılın¸c-Karzan, K¨u¸c¨ukyavuz, and Lee: Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications Then, after subtracting 1 from each side of (21) and taking the square, we obtain(¯ x − ≥ X i ∈ I , α i , | x ,i | + X j ∈ I , α j , q x ,j + 2 = X i ∈ I , α i , | x ,i | + 2 X i ∈ I , α i , | x ,i | X j ∈ I , α j , q x ,j + 2+ X j ∈ I , ( α j , ) ( x ,j + 2) + X j,k ∈ I , : j = k α j , α k , q x ,j + 2 q x ,k + 2 > X i ∈ I , α i , | x ,i | + 2 X i ∈ I , α i , | x ,i | X j ∈ I , α j , | x ,j | + X j ∈ I , ( α j , ) ( x ,j + 2) + X j,k ∈ I , : j = k α j , α k , q x ,j + 2 q x ,k + 2 ≥ X i ∈ I , α i , | x ,i | + 2 X i ∈ I , α i , | x ,i | X j ∈ I , α j , | x ,j | + X j ∈ I , α j , | x ,j | + 2 X j ∈ I , α j , = X i ∈ I , α i , | x ,i | + X j ∈ I , α j , | x ,j | + 12 (22)where the second and third inequalities follow from q x ,j + 2 > | x ,j | , and q x ,j + 2 q x ,k + 2 ≥ | x ,j x ,k | + 2 . Moreover, note that as ¯ x = 1 + p ¯ x + 1 / x = P i ∈ I , α i , x ,i + P j ∈ I , α j , x ,j , we have(¯ x − = ¯ x + 1 / X i ∈ I , α i , x ,i + X j ∈ I , α j , x ,j + 12 . Therefore, we deduce a contradiction from (22) because for convex combination weights α ≥ x − = X i ∈ I , α i , x ,i + X j ∈ I , α j , x ,j + 12 > X i ∈ I , α i , | x ,i | + X j ∈ I , α j , | x ,j | + 12 . This in turn implies that (15) has an extreme point with a fractional zz