A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints
PPreprint, Faculty 1Institute of Mathematics
A new elementary proof forM-stationarity under MPCC-GCQ for mathematical programs withcomplementarity constraints
Felix HarderNovember 6, 2020Chair of Optimal Control, BTU Cottbus-Senftenberg
Abstract
It is known in the literature that local minimizers of mathematical programswith complementarity constraints (MPCCs) are so-called M-stationary points, if avery weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ)holds. In this paper we present a new elementary proof for this result. Our proofis significantly simpler than existing proofs and does not rely on deeper technicaltheory such as calculus rules for limiting normal cones. A crucial ingredient is aproof of a (to the best of our knowledge previously open) conjecture, which wasformulated in a Diploma thesis by Schinabeck.
Keywords:
Mathematical program with complementarity constraints, Mathematicalprogram with equilibrium constraints, Necessary optimality conditions, M-stationarity,Guignard constraint qualification
MSC (2020):
We consider mathematical programs with complementarity constraints, or MPCCs forshort, which are nonlinear optimization problems of the formmin x f ( x )s.t. g ( x ) ≤ , h ( x ) = 0 ,G ( x ) ≥ , H ( x ) ≥ , G ( x ) > H ( x ) = 0 . (MPCC)1 a r X i v : . [ m a t h . O C ] N ov new elementary proof for M-stationarity Felix Harder Here, f : R n → R , g : R n → R l , h : R n → R m , G, H : R n → R p are differentiablefunctions. In the literature, MPCCs are also often called mathematical programs withequilibrium constraints, or MPECs for short.MPCCs have been studied extensively in the literature, both from a numerical and atheoretical perspective. It is well known that standard constraint qualifications are usuallynot satisfied for (MPCC). Therefore, one usually considers constraint qualifications andstationarity conditions that are tailored to MPCCs. One such (necessary, first-order)stationarity condition is strong stationarity, but there are examples of MPCCs where thedata is linear, but the unique minimizer is not strongly stationary, see [Scheel, Scholtes,2000, Example 3].The next strongest stationarity condition for MPCCs in the literature is M-stationarity,which is defined in Definition 2.1. It has been shown in [Flegel, Kanzow, Outrata, 2006]that M-stationarity holds under MPCC-GCQ, see also [Flegel, Kanzow, 2006]. MPCC-GCQ, which is defined in Definition 2.3, is, to the best of our knowledge, the weakestconstraint qualification that is used for MPCCs in the literature. These proofs for M-stationarity under MPCC-GCQ rely on the concept of so-called limiting normal cones. Inparticular, calculus rules for the limiting normal cone are used, which are based on deepertechnical theory and require to verify the calmness of certain set-valued mappings. Thereare also proofs of M-stationarity using nonsmooth regularization methods, see [Kanzow,Schwartz, 2013]. However, they require significantly stronger constraint qualifications.In this paper we want to present a new proof for M-stationarity under MPCC-GCQwhich is elementary, i.e. we do not rely on advanced theory such as the properties oflimiting normal cones. Our proof is significantly simpler than any existing proofs that weare aware of. The major novel contribution of this paper is a result which can be foundin Lemma 3.2. This result was already conjectured in [Schinabeck, 2009, Section 4.4.2],and, to the best of our knowledge there has not been a proof of this conjecture so far.With the knowledge that Lemma 3.2 holds, the rest of the proof of M-stationarity underMPCC-GCQ will not be particularly surprising for readers familiar with the implicationsof MPCC-GCQ. For the convenience of the reader we give a self-contained presentation,which only requires familiarity with basic theory of nonlinear optimization.We hope that the reader gains new insights into the structure of stationarity conditionsfor MPCCs and that this paper makes it easier to fully understand why M-stationarityholds under MPCC-GCQ.The structure of this paper is as follows: In Section 2, we give the relevant definitions.Then we use MPCC-GCQ to construct various multipliers that satisfy an A-stationarysystem in Proposition 3.1. Afterwards, these multipliers are combined into a multiplierwhich is M-stationary with the help of Lemma 3.2. The main result is then stated inTheorem 3.3. Finally, we give a brief outlook and conclusion in Section 4.2 new elementary proof for M-stationarity Felix Harder It will be convenient to work with the index sets I l := { , . . . , l } ,I m := { , . . . , m } ,I p := { , . . . , p } ,I g (¯ x ) := { i ∈ I l | g i (¯ x ) = 0 } ,I +0 (¯ x ) := { i ∈ I p | G i (¯ x ) > ∧ H i (¯ x ) = 0 } ,I (¯ x ) := { i ∈ I p | G i (¯ x ) = 0 ∧ H i (¯ x ) > } ,I (¯ x ) := { i ∈ I p | G i (¯ x ) = 0 ∧ H i (¯ x ) = 0 } , where ¯ x ∈ R n is a feasible point of (MPCC). Note that I +0 (¯ x ) , I (¯ x ) , I (¯ x ) form apartition of I p . We continue with the definition of M- and A-stationarity. Definition 2.1.
Let ¯ x ∈ R n be a feasible point of (MPCC). We call ¯ x an M-stationary point of (MPCC) if there exist multipliers ¯ λ ∈ R l , ¯ η ∈ R m , ¯ µ, ¯ ν ∈ R p with ∇ f (¯ x ) + g (¯ x ) > ¯ λ + h (¯ x ) > ¯ η + G (¯ x ) > ¯ µ + H (¯ x ) > ¯ ν = 0 , (2.1a)¯ λ ≥ , ¯ λ i = 0 ∀ i ∈ I l \ I g (¯ x ) , (2.1b)¯ µ i = 0 ∀ i ∈ I +0 (¯ x ) , (2.1c)¯ ν i = 0 ∀ i ∈ I (¯ x ) , (2.1d)(¯ µ i < ∧ ¯ ν i < ∨ ¯ µ i ¯ ν i = 0 ∀ i ∈ I (¯ x ) . (2.1e)If the multipliers ¯ λ, ¯ η, ¯ µ, ¯ ν only satisfy (2.1a)–(2.1d) and ¯ µ i ≤ ∨ ¯ ν i ≤ i ∈ I (¯ x ), then ¯ x is called an A-stationary point of (MPCC).Other stationarity concepts for (MPCC) can be found in [Ye, 2005, Definitions 2.2–2.7].In preparation for the definition of MPCC-GCQ we introduce some additional concepts.
Definition 2.2.
Let ¯ x ∈ R n be a feasible point of (MPCC).(a) We define the tangent cone of (MPCC) at ¯ x via T (¯ x ) := ( d ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∃{ x k } k ∈ N ⊂ F, ∃{ t k } k ∈ N ⊂ (0 , ∞ ) : x k → ¯ x, t k ↓ , t − k ( x k − ¯ x ) → d ) , where F ⊂ R n denotes the feasible set of (MPCC).3 new elementary proof for M-stationarity Felix Harder (b) We define the MPCC-linearized tangent cone T linMPCC (¯ x ) ⊂ R n at ¯ x via T linMPCC (¯ x ) := d ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ g i (¯ x ) > d ≤ ∀ i ∈ I g (¯ x ) , ∇ h i (¯ x ) > d = 0 ∀ i ∈ I m , ∇ G i (¯ x ) > d = 0 ∀ i ∈ I (¯ x ) , ∇ H i (¯ x ) > d = 0 ∀ i ∈ I +0 (¯ x ) , ∇ G i (¯ x ) > d ≥ ∀ i ∈ I (¯ x ) , ∇ H i (¯ x ) > d ≥ ∀ i ∈ I (¯ x ) , ( ∇ G i (¯ x ) > d )( ∇ H i (¯ x ) > d ) = 0 ∀ i ∈ I (¯ x ) . (c) For a set C ⊂ R n its polar cone C ◦ ⊂ R n is defined via C ◦ := { d ∈ R n | d > y ≤ ∀ y ∈ C } . Note that, in general, T (¯ x ) and T linMPCC (¯ x ) are nonconvex sets.Now we are ready to give the definition of MPCC-GCQ, which can also be found in[Flegel, Kanzow, Outrata, 2006, (41)], where it is called MPEC-GCQ. Definition 2.3.
Let ¯ x ∈ R n be a feasible point of (MPCC). We say that ¯ x satisfiesthe MPCC-tailored Guignard constraint qualification , or
MPCC-GCQ , if T (¯ x ) ◦ = T linMPCC (¯ x ) ◦ holds. Additionally, if T (¯ x ) = T linMPCC (¯ x ) holds then we say that ¯ x satisfies MPCC-ACQ .Clearly, MPCC-ACQ implies MPCC-GCQ. We mention that there are also other strongerconstraint qualifications (such as MPCC-MFCQ if g , h , G , H are continuously differen-tiable) which imply MPCC-ACQ or MPCC-GCQ and are easier to verify, see e.g. [Ye,2005, Theorem 3.2]. In particular, we emphasize that MPCC-GCQ (and MPCC-ACQ)are satisfied at every feasible point of (MPCC) if the functions g , h , G , H are affine. We start with a proposition that generates several multipliers which satisfy a slightlystronger stationarity condition than A-stationarity. The result can also be obtained fromthe proof of [Flegel, Kanzow, 2005, Theorem 3.4], with the minor difference that we onlyrequire MPCC-GCQ and not MPCC-ACQ.
Proposition 3.1.
Let ¯ x ∈ R n be a local minimizer of (MPCC) that satisfies MPCC-GCQ and let α ∈ { , } p be given. Then there exist multipliers λ α ∈ R l , η α ∈ R m ,4 new elementary proof for M-stationarity Felix Harder µ α , ν α ∈ R p with ∇ f (¯ x ) + g (¯ x ) > λ α + h (¯ x ) > η α + G (¯ x ) > µ α + H (¯ x ) > ν α = 0 , (3.1a) λ α ≥ , λ αi = 0 ∀ i ∈ I l \ I g (¯ x ) , (3.1b) µ αi = 0 ∀ i ∈ I +0 (¯ x ) , (3.1c) ν αi = 0 ∀ i ∈ I (¯ x ) , (3.1d) α i = 1 ⇒ µ αi ≤ ∀ i ∈ I (¯ x ) , (3.1e) α i = 2 ⇒ ν αi ≤ ∀ i ∈ I (¯ x ) . (3.1f) Proof.
With Definition 2.2 (a) and the fact that ¯ x is a local minimizer of (MPCC) it iseasy to see that the condition ∇ f (¯ x ) > d ≥ ∀ d ∈ T (¯ x )is satisfied. Using polar cones and MPCC-GCQ, we obtain −∇ f (¯ x ) ∈ T (¯ x ) ◦ = T linMPCC (¯ x ) ◦ ⊂ T linNLP( α ) (¯ x ) ◦ , where the cone T linNLP( α ) (¯ x ) ⊂ T linMPCC (¯ x ) is defined via T linNLP( α ) (¯ x ) := ( d ∈ T linMPCC (¯ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α i = 1 ⇒ ∇ H i (¯ x ) > d = 0 ∀ i ∈ I (¯ x ) ,α i = 2 ⇒ ∇ G i (¯ x ) > d = 0 ∀ i ∈ I (¯ x ) ) . Note that, unlike T linMPCC (¯ x ), this is a convex and polyhedral cone. Thus, one cancalculate its polar cone (e.g. using Farkas’ Lemma), which results in T linNLP( α ) (¯ x ) ◦ = X i ∈ I g (¯ x ) λ αi ∇ g i (¯ x ) + X i ∈ I m η αi ∇ h i (¯ x )+ X i ∈ I (¯ x ) ∪ I (¯ x ) µ αi ∇ G i (¯ x )+ X i ∈ I +0 (¯ x ) ∪ I (¯ x ) ν αi ∇ H i (¯ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ αi ≥ , i ∈ I g (¯ x ) ,η αi ∈ R , i ∈ I m ,µ αi ∈ R , i ∈ I (¯ x ) ∪ I (¯ x ) ,ν αi ∈ R , i ∈ I +0 (¯ x ) ∪ I (¯ x ) ,µ αi ≤ , if α i = 1 , i ∈ I (¯ x ) ,ν αi ≤ , if α i = 2 , i ∈ I (¯ x ) . Then the result follows from −∇ f (¯ x ) ∈ T linNLP( α ) (¯ x ) ◦ by setting the remaining compo-nents of the multipliers (i.e. λ αi for i ∈ I p \ I g (¯ x ), µ αi for i ∈ I +0 (¯ x ), ν αi for i ∈ I (¯ x ))to zero.Clearly, if ¯ x satisfies (3.1) for some α ∈ { , } p and suitable multipliers, then ¯ x is anA-stationary point of (MPCC). However, the statement of Proposition 3.1 is strongerthan A-stationarity, namely for each index in I (¯ x ) we can choose whether µ αi or ν αi isnon-positive. Note that (2.1a)–(2.1d) are already satisfied by all 2 p possible choices for5 new elementary proof for M-stationarity Felix Harder the multipliers and any convex combination of these. Thus, the question naturally ariseswhether a convex combination of these multipliers can be found that also satisfies (2.1e).As the next result shows, this is indeed possible. The following lemma was already statedas a conjecture in [Schinabeck, 2009, Section 4.4.2]. To the best of our knowledge, thisconjecture has not been proven before. Lemma 3.2.
Let ˆ I ⊂ I p be an index set. Suppose that for all α ∈ { , } p there existpoints ( µ α , ν α ) ∈ A α , where A α := { ( µ, ν ) ∈ R p | µ i ≤ α i = 1 , ν i ≤ α i = 2 ∀ i ∈ ˆ I } . Then we can find a point (¯ µ, ¯ ν ) in the set B := conv (cid:8) ( µ α , ν α ) (cid:12)(cid:12) α ∈ { , } p (cid:9) ⊂ R p of convex combinations of these points, such that for all i ∈ ˆ I we have the condition(¯ µ i < ∧ ¯ ν i < ∨ ¯ µ i ¯ ν i = 0 . (3.2) Proof.
Let us choose points (ˆ µ α , ˆ ν α ) ∈ B ∩ A α , ¯ µ, ¯ ν ∈ R p and a vector β ∈ { , } p thatsatisfy (ˆ µ α , ˆ ν α ) ∈ arg min ( µ,ν ) ∈ B ∩ A α k ( µ, ν ) k ∀ α ∈ { , } p , (3.3) β ∈ arg max α ∈{ , } p k (ˆ µ α , ˆ ν α ) k , (3.4)(¯ µ, ¯ ν ) := (ˆ µ β , ˆ ν β ) ∈ R p . (3.5)Clearly, these choices are possible. Furthermore, we have (¯ µ, ¯ ν ) ∈ B , i.e. it is a convexcombination as claimed.Let i ∈ ˆ I be given. It remains to show that our choice for (¯ µ, ¯ ν ) satisfies (3.2). Withoutloss of generality we can assume that β i = 1 holds (otherwise one would exchange theroles of µ and ν in the rest of the proof). Therefore, we have ¯ µ i ≤ µ, ¯ ν ) ∈ A β .Suppose that (3.2) is not satisfied, i.e. ¯ µ i < ν i > γ ∈ { , } p , γ j := ( j = i,β j if j ∈ I p \ { i } ∀ j ∈ I p . Due to ¯ µ i < t ∈ (0 ,
1) such that the convex combination( µ t , ν t ) := t (ˆ µ γ , ˆ ν γ ) + (1 − t )(¯ µ, ¯ ν ) ∈ R p still satisfies µ ti <
0. Since γ j = β j holds for j = i we also have ( µ t , ν t ) ∈ A β . However,(¯ µ, ¯ ν ) = (ˆ µ γ , ˆ ν γ ) due to ˆ ν γi ≤
0, i.e. ( µ t , ν t ) is a strict convex combination. Thus, byalso using (3.4), we have k ( µ t , ν t ) k < max (cid:8) k (¯ µ, ¯ ν ) k , k (ˆ µ γ , ˆ ν γ ) k (cid:9) ≤ k (ˆ µ β , ˆ ν β ) k . Due to ( µ t , ν t ) ∈ B ∩ A β this is a contradiction to (3.3), which completes the proof.6 new elementary proof for M-stationarity Felix Harder We mention that it was recognized already in [Schinabeck, 2009, Section 4.4.2] that thislemma would significantly simplify the already existing proofs for M-stationarity.A straightforward combination of Proposition 3.1 and Lemma 3.2 yields the desiredM-stationarity result.
Theorem 3.3.
Let ¯ x ∈ R n be a local minimizer of (MPCC) that satisfies MPCC-GCQ.Then ¯ x is an M-stationary point. Proof.
For all α ∈ { , } p , let ( λ α , η α , µ α , ν α ) ∈ R l + m +2 p be the multipliers generated byProposition 3.1. By applying Lemma 3.2 with ˆ I = I (¯ x ), we find a convex combination(¯ λ, ¯ η, ¯ µ, ¯ ν ) ∈ R l + m +2 p of these multipliers such that (2.1e) is satisfied. The conditions(2.1a)–(2.1d) follow from (3.1a)–(3.1d) by convexity. We provided an new proof for M-stationarity of local minimizers of MPCCs underMPCC-GCQ. Although this result was already known, the new proof uses only basic andwell-known tools from the theory for nonlinear programming. This new elementary prooffor M-stationarity was enabled by providing a proof for a (to the best of our knowledgepreviously open) conjecture from [Schinabeck, 2009] in Lemma 3.2.In the future, it would also be interesting to apply this approach to other problem classesfrom disjunctive programming and to investigate to what extend the ideas from thispaper can be generalized.In Sobolev or Lebesgue spaces, the limiting normal cone turned out to be not as effectiveas in finite dimensional spaces for obtaining stationarity conditions for complementarity-type optimization problems, see [Harder, Wachsmuth, 2018; Mehlitz, Wachsmuth, 2018].Thus, it would be interesting to know whether the new elementary method from thispaper can provide ideas for possible approaches for better stationarity conditions ofcomplementarity-type optimization problems in Sobolev and Lebesgue spaces. However,it would not be trivial to transfer the method from finite-dimensional spaces to infinite-dimensional spaces.
Acknowledgments
The author would like to thank Gerd Wachsmuth for making the authoraware of the conjecture in the Diploma thesis [Schinabeck, 2009]. This work is supported bythe DFG Grant
Bilevel Optimal Control: Theory, Algorithms, and Applications (Grant No. WA3636/4-2) within the Priority Program SPP 1962 (Non-smooth and Complementarity-basedDistributed Parameter Systems: Simulation and Hierarchical Optimization). new elementary proof for M-stationarity Felix Harder References
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