Portfolio optimization with two quasiconvex risk measures
aa r X i v : . [ q -f i n . P M ] D ec Portfolio optimization with two quasiconvex risk measures
C¸ a˘gın Ararat ∗ December 11, 2020
Abstract
We study a static portfolio optimization problem with two risk measures: a principle riskmeasure in the objective function and a secondary risk measure whose value is controlled in theconstraints. This problem is of interest when it is necessary to consider the risk preferences oftwo parties, such as a portfolio manager and a regulator, at the same time. A special case ofthis problem where the risk measures are assumed to be coherent (positively homogeneous) isstudied recently in a joint work of the author. The present paper extends the analysis to a moregeneral setting by assuming that the two risk measures are only quasiconvex. First, we studythe case where the principal risk measure is convex. We introduce a dual problem, show thatthere is zero duality gap between the portfolio optimization problem and the dual problem, andfinally identify a condition under which the Lagrange multiplier associated to the dual problemat optimality gives an optimal portfolio. Next, we study the general case without the convexityassumption and show that an approximately optimal solution with prescribed optimality gapcan be achieved by using the well-known bisection algorithm combined with a duality resultthat we prove.
Keywords and phrases: portfolio optimization, quasiconvex risk measure, minimal penaltyfunction, maximal risk function, Lagrange duality, bisection method
Mathematics Subject Classification (2010):
Risk measures are functionals that are defined on a linear space of real-valued random variableshosted by a common probability space. In the context of financial mathematics, each randomvariable can denote the uncertain future worth of an investor’s position, and a risk measure assignsa (deterministic) extended real number to the random variable; this number quantifies the initialcapital that is needed for compensating the risk of the position.Introduced in Artzner et al. (1999), risk measures have been studied extensively in the financialmathematics literature over the last two decades. In Artzner et al. (1999), the so-called coherentrisk measures are studied within an axiomatic framework; we provide mathematical formulationsof these axioms in Section 2 for the convenience of the reader. Among the properties of a coherentrisk measure, positive homogeneity imposes that the risk of a financial position is scalable by thesize of the position. While some classical risk measures such as negative expected value and averagevalue-at-risk (see F¨ollmer, Schied (2016, Example 4.40)) enjoy this property, positive homogeneitycan be found restrictive from a financial point of view. To this end, convex risk measures providea richer class of risk measures where positive homogeneity is not taken for granted. A classical ∗ Bilkent University, Department of Industrial Engineering, Ankara, Turkey, [email protected]. xample of a convex but not coherent risk measure is the entropic risk measure, which has a simpleexpression of the log-sum-exp form (see Example 6.1). The reader is referred to F¨ollmer, Schied(2016, Chapter 4) for a detailed discussion on convex risk measures.The convexity property of a risk measure is often motivated by the statement “Diversificationdoes not increase risk,” which emphasizes the role of allocating one’s capital into a variety ofinvestment opportunities. More recently, it has been argued that quasiconvexity can be used asa relaxed alternative for convexity as it still captures the idea behind diversification. Therefore, quasiconvex risk measures , as argued in Cerreia-Vioglio et al. (2011) and Drapeau, Kupper (2013),cover a wider range of functionals that can be used for risk measurement purposes; these includecertainty equivalents (see Drapeau, Kupper (2013, Example 8) and Section 6.2) and economicindices of riskiness (see Drapeau, Kupper (2013, Example 3)) in addition to convex (and coherent)risk measures described above.A rich class of problems where risk measures appear naturally is that of portfolio optimizationproblems. In these problems, one wishes to minimize or control the risk of the future value of aportfolio that consists of multiple risky assets. For special families of asset return distributions,the works Landsman (2008); Landsman, Makov (2016); Owadally (2011) study static portfoliooptimization problems with a single coherent risk measure that appears in the objective function.More recently, in the previous joint work Akt¨urk, Ararat (2020) of the author, a static portfoliooptimization problem with two coherent risk measures is formulated. In this problem, the decision-maker aims to minimize the value of a principle risk measure, e.g., the risk measure of the portfoliomanager, while keeping the value of a secondary risk measure, e.g., the risk measure declared by aregulatory authority, below a critical threshold. In Akt¨urk, Ararat (2020), a complete analysis ofthis problem is provided for the general case of arbitrary asset return distributions and arbitrarycoherent risk measures that satisfy certain regularity conditions.On the other hand, the use of quasiconvex risk measures in portfolio optimization is relativelynew. In Mastrogiacomo, Rosazza Gianin (2015), a static portfolio optimization problem is stud-ied, where the objective function is the composition of a quasiconvex risk measure and a concavefunctional that is defined on the space of portfolios, and this composition is to be minimized overa convex compact set of portfolios. The main result Mastrogiacomo, Rosazza Gianin (2015, Theo-rem 4) provides a sufficient condition for a portfolio to be optimal in terms of a set relation betweennormal cones and generalized subdifferentials for quasiconvex functions. In particular, the deriva-tions rely on the dual representations for quasiconvex risk measures developed in Drapeau, Kupper(2013) as well as the general duality theory for quasiconvex functions initiated earlier in Penot, Volle(1990). In K¨allblad (2017), quasiconvex risk measures are used in a dynamic portfolio optimizationproblem in continuous time in order to model ambiguity-averse preferences. The work K¨allblad(2017) also makes use of the dual representation results of Drapeau, Kupper (2013).The aim of the present paper is to extend the static portfolio optimization problem in Akt¨urk, Ararat(2020) by assuming that both the principle and the secondary risk measures are quasiconvex. Inparticular, we cover the case where the two risk measures are convex. It should be noted that theextension from the coherent case to the quasiconvex case requires entirely different duality argu-ments, explaining the mathematical originality of the present paper. On the other hand, comparedto the portfolio optimization problem in Mastrogiacomo, Rosazza Gianin (2015) with a single qua-siconvex risk measure and a general convex set constraint, our problem assumes that the constrainthas a special structure induced by the secondary risk measure. This structure makes it possible toformulate a more explicit dual problem with a linear inequality constraint.The rest of the paper is organized as follows. After reviewing some basic notions about qua-siconvex risk measures in Section 2 and introducing the primal problem in Section 3, we breakdown the analysis of the problem into two steps. First, in Section 4, we work under the assumption2hat the principle risk measure is a convex functional (not necessarily translative though). Weformulate the dual problem in Section 4.1 and prove that (Theorem 4.8) there is zero duality gapbetween the primal and dual problems. In Section 4.2, we impose further structural propertieson the principle risk measure and prove that (Theorem 4.11) a Lagrange multiplier attached to alinear inequality constraint of the dual problem at optimality yields an optimal portfolio vector forthe primal problem. Next, in Section 5, we remove the convexity assumption and reformulate thequasiconvex portfolio optimization problem via a family of convex feasibility problems parametrizedby a decision variable of the quasiconvex problem. Similar to the results of Section 4, we provide aduality-based method to solve each of these feasibility problems. Then, we employ the well-knownbisection method that iterates through different values of the parameter of the feasibility problemsand stops with prescribed suboptimality in finitely many iterations. Hence, combining the dualityresult with the bisection method provides a way to find an approximately optimal solution for theportfolio optimization problem under quasiconvex risk measures. Finally, in Section 6, we considerconvex risk measures and certainty equivalents in order to illustrate the use of the dual problem,and we discuss the validity of some technical assumptions stated in Section 4 and Section 5.
In this section, we fix the notation for the rest of the paper and review some preliminary notionsrelated to risk measures. For the latter, we focus on the more recent quasiconvex framework studiedin Cerreia-Vioglio et al. (2011); Drapeau, Kupper (2013).Let n ∈ N := { , , . . . } . We assume that the standard Euclidean space R n is equipped withan arbitrary norm |·| and the usual inner product defined by x T z := P ni =1 x i z i for x, z ∈ R n . Wedenote by R n + the positive orthant in R n , that is, the set of all x = ( x , . . . , x n ) T ∈ R n with x i ≥ i ∈ { , . . . , n } .To introduce the probabilistic setup, let (Ω , F , P ) be a probability space and denote by L n theset of all F -measurable random variables taking values in R n , where two elements are consideredidentical if they are equal P -almost surely. For each X ∈ L n , we define k X k p := ( E [ | X | p ]) p for p ∈ [1 , + ∞ ) and k X k p := inf { c ≥ | P {| X | ≤ c } = 1 } for p = + ∞ . Let p ∈ [1 , + ∞ ]. The space L pn := { X ∈ L n | k X k p < + ∞} is a Banach spaceequipped with the norm k·k p . For brevity, let L p := L p ; for Y , Y ∈ L p , we write Y ≤ Y if P { Y ≤ Y } = 1, which yields the cone L p + := { Y ∈ L p | ≤ Y } .Let Y = L p , where p ∈ [1 , + ∞ ]. The space Y is considered with its strong topology inducedby the norm k·k p if p < + ∞ and with the weak ∗ topology σ ( L ∞ , L ) if p = + ∞ . Under thistopology, we denote by Y ∗ the topological dual space of Y with the bilinear duality mapping h· , ·i : Y ∗ × Y → R . Hence, Y ∗ = L q with h V, Y i = E [ V Y ] for every V ∈ Y ∗ , Y ∈ Y , where q ∈ [1 , + ∞ ] is the conjugate exponent of p , that is, p + q = 1. In all cases, we consider Y ∗ withthe weak topology σ ( Y ∗ , Y ).For a functional ρ : Y → ¯ R := [ −∞ , + ∞ ], let us consider the following properties. (i) Monotonicity: Y ≤ Y implies ρ ( Y ) ≥ ρ ( Y ) for every Y , Y ∈ Y . (ii) Quasiconvexity: It holds ρ ( λY + (1 − λ ) Y ) ≤ max { ρ ( Y ) , ρ ( Y ) } for every Y , Y ∈ Y and λ ∈ (0 , iii) Convexity: It holds ρ ( λY + (1 − λ ) Y ) ≤ λρ ( Y ) + (1 − λ ) ρ ( Y ) for every Y , Y ∈ Y and λ ∈ (0 ,
1) (with the inf-addition convention (+ ∞ ) + ( −∞ ) = + ∞ for the right hand side). (iv) Translativity: It holds ρ ( Y + y ) = ρ ( Y ) − y for every Y ∈ Y and y ∈ R . (v) Positive homogeneity: It holds ρ ( λY ) = λρ ( Y ) for every Y ∈ Y and λ ≥ t ∈ R , let us define A t := { Y ∈ Y | ρ ( Y ) ≤ t } , which is called the acceptance set of ρ at level t . It is easy to check that ρ ( Y ) = inf (cid:8) t ∈ R | Y ∈ A t (cid:9) , Y ∈ Y . (2.1)Note that ρ satisfies quasiconvexity if and only if A t is convex for each t ∈ R . The functional ρ is called a quasiconvex risk measure if it satisfies (i) and (ii), a convex risk measure if it satisfies(i), (ii), (iii), (iv), and a coherent risk measure if it satisfies (i), (ii), (iii), (iv), (v). Hence, acoherent risk measure is necessarily a convex risk measure, and a convex risk measure is necessarilya quasiconvex risk measure. In addition, convexity implies quasiconvexity; and, under monotonicity,quasiconvexity and translativity imply convexity; see F¨ollmer, Schied (2016, Exercise 4.1.1). Hence,when working within the general framework of quasiconvex risk measures, one does not assumetranslativity.In the current paper, we study a static portfolio optimization problem with two quasiconvexrisk measures. In order to formulate a dual problem associated to the portfolio optimizationproblem, the dual representations of these risk measures will have a crucial role. We review thedual representation result of Drapeau, Kupper (2013) next. To that end, let Y ∗ + := L q + . Definition 2.1. (Drapeau, Kupper, 2013, Definition 9) A function β : ( Y ∗ + \{ } ) × R → ¯ R is calleda maximal risk function if it satisfies the following properties.(a) β is increasing and left-continuous in the second argument.(b) β is jointly quasiconcave.(c) It holds β ( λV, λs ) = β ( V, s ) for every V ∈ Y ∗ + , s ∈ R , λ > .(d) It holds lim s →−∞ β ( V , s ) = lim s →−∞ β ( V , s ) for every V , V ∈ Y ∗ + .(e) The right-continuous version ( V, s ) β + ( V, s ) := inf s ′ >s β ( V, s ′ ) of β is (weakly) upper semi-continuous in the first argument. Thanks to property (c) in Definition 2.1, one can simply work with the restriction of β on theset Y ∗ , × R , where Y ∗ , := L q, := (cid:8) Y ∈ L q + | E [ Y ] = 1 (cid:9) . (2.2)Let ρ be a lower semicontinuous quasiconvex risk measure. We define the minimal penaltyfunction α : Y ∗ + × R → ¯ R of ρ by α ( V, t ) := sup Y ∈ A t E [ − V Y ] , V ∈ Y ∗ + , t ∈ R . For each t ∈ R , the acceptance set A t is a closed convex subset of Y , hence it is characterizedby its support function V α ( V, t ). Consequently, the risk measure ρ is uniquely determined by4ts minimal penalty function α . In quasiconvex analysis, it is sometimes useful to work with theleft-continuous version α − of α (with respect to the second variable) defined by α − ( V, t ) := sup t ′ Assumption 4.3. For every w ∈ W , V ∈ L q, and V ∈ L q + , it holds β ( V , E [ − V w T X ]) ∈ R and α ( V , r ) ∈ R . The next result establishes the connection between ( P ( r )) and h P . Proposition 4.4. Suppose that Assumption 4.1, Assumption 4.2, Assumption 4.3 hold. Then, p ( r ) = sup V ,V ∈ L q + (cid:0) h P ( V , V ) − α ( V , r ) (cid:1) . Proof. Let I A r be the convex analytic indicator function of A r , that is, I A r ( Y ) = 0 whenever Y ∈ A r and I A r ( Y ) = + ∞ whenever Y ∈ L p \ A r . Then, using Z˘alinescu (2002, Theorem 2.3.3,(2.33)), Proposition 2.2 and property (c) in Definition 2.1, we obtain p ( r ) = inf n ρ ( w T X ) | ρ ( w T X ) ≤ r, w ∈ W o = inf n ρ ( w T X ) | w T X ∈ A r , w ∈ W o = inf w ∈W (cid:16) ρ ( w T X ) + I A r ( w T X ) (cid:17) = inf w ∈W sup V ∈ L q, β (cid:16) V , E [ − V w T X ] (cid:17) + sup V ∈ L q + (cid:16) E [ − V w T X ] − α ( V , r ) (cid:17) = inf w ∈W sup V ∈ L q, ,V ∈ L q + b ( w, V , V ) , where b ( w, V , V ) := β (cid:16) V , E [ − V w T X ] (cid:17) + E [ − V w T X ] − α ( V , r ) . For fixed w ∈ W , V ∈ L q, , clearly V E [ − V w T X ] − α ( V , r ) is concave and upper semicon-tinuous by the properties of support function. Let w ∈ W , V ∈ L q + . For each V , V ′ ∈ L q, and λ ∈ (0 , β (cid:16) λV + (1 − λ ) V ′ , E [ − ( λV + (1 − λ ) V ′ ) w T X ] (cid:17) = β (cid:16) λV + (1 − λ ) V ′ , λ E [ − V w T X ] + (1 − λ ) E [ − V ′ w T X ] (cid:17) ≥ min n β (cid:16) V , E [ − V w T X ] (cid:17) , β (cid:16) V ′ , E [ − V ′ w T X ] (cid:17)o 7y the joint quasiconcavity of β . Hence, the function V β (cid:0) V , E [ − V w T X ] (cid:1) is quasiconcave.We claim that this function is also weakly upper semicontinuous. To that end, let ( V α ) α ∈ I be aweakly convergent net in L q, with some index set I and limit V . Hence, lim α →∞ E [ − V α w T X ] = E [ − V w T X ]. So ( V α , E [ − V α w T X ]) α ∈ I is a weakly convergent net in L q, × R with limit ( V, E [ − V w T X ]).By Assumption 4.2, we getlim sup α →∞ β ( V α , E [ − V α w T X ]) ≤ β ( V, E [ − V w T X ]) . Hence, the claim follows. Since β is increasing and left-continuous in the second argument, it is alsolower semicontinuous (indeed continuous by Assumption 4.2). Hence, for fixed V ∈ L q, , V ∈ L q + ,the function w β (cid:0) V , E [ − V w T X ] (cid:1) is lower semicontinuous (indeed continuous) and quasicon-vex. Note that W is a convex and compact set. Finally, by Assumption 4.3, b is real-valued on W × L q, × L q + . Therefore, by Sion’s minimax theorem (Sion, 1958, Corollary 3.3), we may write p ( r ) = sup V ∈ L q, ,V ∈ L q + inf w ∈W b ( w, V , V )= sup V ∈ L q, ,V ∈ L q + (cid:18) inf w ∈W (cid:16) β ( V , E [ − V X ] T w ) + E [ − V X ] T w (cid:17) − α ( V , r ) (cid:19) = sup V ,V ∈ L q + (cid:18) inf w ∈W (cid:16) β ( V , E [ − V X ] T w ) + E [ − V X ] T w (cid:17) − α ( V , r ) (cid:19) = sup V ∈ V ∈ L q + (cid:18) inf w ∈W (cid:16) inf n t ∈ R | α − ( V , t ) ≥ E [ − V X ] T w o + E [ − V X ] T w (cid:17) − α ( V , r ) (cid:19) = sup V ,V ∈ L q + (cid:0) h P ( V , V ) − α ( V , r ) (cid:1) , where h P ( V , V ) is defined by (4.1). Remark 4.5. In the proof of Proposition 4.4, the only role of Assumption 4.3 is to ensure thatthe function b has finite values so that Sion’s minimax theorem can be applied. In cases whereAssumption 4.3 is not valid, the proof of Proposition 4.4 can be seen as a heuristic argument tocome up with a dual formulation of ( p ( r )). Thanks to Theorem 4.11 below, it turns out that theconclusion of Proposition 4.4 is still valid without Assumption 4.3.Let V , V ∈ L q + . Note that h P ( V , V ) is the optimal value of a finite-dimensional optimizationproblem which is in general nonconvex due to the inequality constraint when ρ is only assumedto be a quasiconvex risk measure. However, under Assumption 4.1, as argued in the proof ofDrapeau, Kupper (2013, Proposition 6), the function t α − ( V , t ) is concave and the optimizationproblem in h P ( V , V ) becomes convex. Before proceeding further, we introduce an additionalassumption related to the asymptotic behavior of α − . Assumption 4.6. For each V ∈ L q + , it holds lim t →∞ α − ( V , t ) = + ∞ . To formulate the next proposition, let us define α ∗ ( V , z ) := inf t ∈ R (cid:0) tz − α − ( V , t ) (cid:1) = inf t ∈ R ( tz − α ( V , t )) , V ∈ L q + , z ∈ R . z 7→ − α ∗ ( V , − z ) is the conjugate function of α − (and also of α ) with respect to thesecond variable. Since t α − ( V , t ) is an increasing function, it is easy to check that α ∗ ( V , z ) = −∞ whenever z < 0. Moreover, under Assumption 4.1, for each V ∈ L q + , the function t α − ( V , t ) is concave and upper semicontinuous so that Fenchel-Moreau theorem (Z˘alinescu, 2002,Theorem 2.3.3) gives α − ( V , t ) = inf z ≥ ( tz − α ∗ ( V , z )) , V ∈ L q + , t ∈ R . (4.2)The following special value of α ∗ will play an important role in the dual problem of ( P ( r )):˜ α ( V ) := α ∗ ( V , 1) = inf t ∈ R (cid:0) t − α − ( V , t ) (cid:1) , V ∈ L q + . (4.3)Using this quantity, let us define, for each V , V ∈ L q + , h D ( V , V ) := sup { ˜ α ( xV ) − y | x E [ V X ] + E [ V X ] ≤ y , x > , y ∈ R } , where the inequality constraint is understood in the componentwise manner. As the proof of thenext result shows, the dual of the problem in h P ( V , V ) gives rise to h D ( V , V ). Proposition 4.7. Suppose that Assumption 4.1 and Assumption 4.6 hold, and let V , V ∈ L q + .Then, h P ( V , V ) = h D ( V , V ) . Proof. As noted above, the problem defining h P ( V , V ) is a convex optimization problem underAssumption 4.1. By Assumption 4.6, there exists w ∈ W and t ∈ R such that α − ( V , t ) > E [ − V X ] T w (indeed, for every w ∈ W such t ∈ R exists). Hence, Slater’s condition holds forthis convex optimization problem and we have h P ( V , V ) = sup x ≥ ,y ∈ R inf w ∈ R n + ,t ∈ R : α − ( V ,t ) ∈ R (cid:16) t + E [ − V X ] T w + x ( − α − ( V , t ) + E [ − V X ] T w ) + y ( T w − (cid:17) = sup x ≥ ,y ∈ R inf w ∈ R n + ,t ∈ R : α − ( V ,t ) ∈ R (cid:16) t − xα − ( V , t ) + ( x E [ − V X ] + E [ − V X ] + y ) T w − y (cid:17) = sup x> ,y ∈ R inf w ∈ R n + ,t ∈ R : α − ( V ,t ) ∈ R (cid:16) t − α − ( xV , t ) + ( x E [ − V X ] + E [ − V X ] + y ) T w − y (cid:17) = sup x> ,y ∈ R { ˜ α ( xV ) − y | x E [ V X ] + E [ V X ] ≤ y } . Let us justify each passage in the above derivation: the first equality is by strong duality for convexoptimization, the second is by simple manipulations, the third excludes the case x = 0 from furtherconsideration since in this case the infimum yields −∞ , the fourth is by evaluating the infimumwith respect to w ∈ R n + and t ∈ R . Therefore, h P ( V , V ) = h D ( V , V ).We are ready to prove the first main result of the paper, which establishes strong duality between( P ( r )) and a new problem, which we refer to as the dual problem of ( P ( r )).9 heorem 4.8. Suppose that Assumption 4.1, Assumption 4.2, Assumption 4.3, Assumption 4.6hold, and consider the problem maximize ˜ α ( V ) − α ( V , r ) − y ( D ( r )) subject to E [ V X ] + E [ V X ] ≤ y V , V ∈ L q + , y ∈ R , where the inequality constraint is understood in the componentwise sense. Then, ( P ( r )) and ( D ( r )) have the same optimal value p ( r ) . Proof. Combining Proposition 4.4 and Proposition 4.7, we obtain p ( r ) = sup V ,V ∈ L q + (cid:0) h D ( V , V ) − α ( V , r ) (cid:1) = sup V ,V ∈ L q + (sup { ˜ α ( xV ) − y | E [ xV X ] + E [ V X ] ≤ y , x > , y ∈ R } − α ( V , r ))= sup (cid:8) ˜ α ( xV ) − α ( V , r ) − y | E [ xV X ] + E [ V X ] ≤ y , V , V ∈ L q + , x > , y ∈ R (cid:9) = sup (cid:8) ˜ α ( V ) − α ( V , r ) − y | E [ V X ] + E [ V X ] ≤ y , V , V ∈ L q + , y ∈ R (cid:9) , which coincides with the optimal value of ( D ( r )).It is worth noting that the dual problem ( D ( r )) is a convex optimization problem thanks toAssumption 4.1. While Theorem 4.8 provides strong duality between ( P ( r )) and ( D ( r )), it does not make a state-ment on how to find an optimal portfolio w ∗ ∈ W for ( P ( r )). The aim of Theorem 4.11, the secondmain result of the paper, is to find such w ∗ in relation to ( D ( r )).As a preparation, we recall some well-known concepts and facts from convex analysis. To thatend, let us fix an arbitrary Hausdorff locally convex topological vector space X with topologicaldual X ∗ and bilinear duality mapping h· , ·i : X ∗ × X → R . For our purposes, the following specialcases of X are particularly important:(i) X = R n with the usual topology, which yields X ∗ = R n together with h z, x i = z T x for every x, z ∈ R n .(ii) X = L q with q ∈ [1 , + ∞ ) with the weak topology σ ( L q , L p ), which yields X ∗ = L p togetherwith h Y, U i = E [ U Y ] for every U ∈ L q , Y ∈ L p .(iii) X = L ∞ with the weak topology σ ( L ∞ , L ), which yields X ∗ = L together with h Y, U i = E [ U Y ] for every U ∈ L ∞ , Y ∈ L .Consider a set A ⊆ X . The function I A : X → R ∪ { + ∞} defined by I A ( x ) = ( x ∈ A, + ∞ if x ∈ X \ A, is called the indicator function of A ; note that A is a convex set if and only if I A is a convexfunction. For a point x ∈ A , the convex cone N ( A, x ) := (cid:8) z ∈ X ∗ | ∀ x ′ ∈ A : h z, x i ≥ (cid:10) z, x ′ (cid:11)(cid:9) 10s called the normal cone of A at x . Let g : X → R ∪ { + ∞} be a function. Given x ∈ X , the set ∂g ( x ) := (cid:8) z ∈ X ∗ | ∀ x ′ ∈ X : g ( x ′ ) ≥ g ( x ) + (cid:10) z, x ′ − x (cid:11)(cid:9) is called the subdifferential of g at x . If A is a nonempty convex set, then by Z˘alinescu (2002,Section 2.4), ∂I A ( x ) = N A ( x ) for every x ∈ A , and ∂I A ( x ) = ∅ for every x ∈ X \ A . The function g ∗ : X ∗ → ¯ R defined by g ∗ ( z ) := sup x ∈X ( h z, x i − g ( x )) for each z ∈ X ∗ is called the conjugatefunction of g . We have z ∈ ∂g ( x ) ⇔ x ∈ ∂g ∗ ( z ) (4.4)for every x ∈ X , z ∈ X ∗ such that g is lower semicontinuous at x . If A is a nonempty closed convexset, then it is well-known that σ A := ( I A ) ∗ is the support function of A defined by σ A ( z ) := sup x ∈ A h z, x i , z ∈ X ∗ . From the above definitions, it is clear that, for a point x ∈ A , we have N ( A, x ) = { z ∈ X ∗ | σ A ( z ) = h z, x i} . (4.5)Consider the problem of minimizing g over A . Suppose that g is convex and let x ∈ A with g ( x ) < + ∞ . By Pshenichnyi-Rockafellar theorem (Z˘alinescu, 2002, Theorem 2.9.1), if ∂g ( x ) ∩ −N ( A, x ) = ∅ , (4.6)then x is a minimizer of g over A , that is, g ( x ) = inf x ′ ∈ A g ( x ′ ); the converse also holds if g iscontinuous at x .The next lemma is devoted to the calculation of a certain subdifferential that is relevant to( P ( r )). Lemma 4.9. Suppose that Assumption 4.1 holds. Let w ∈ R n be such that g ( w ) = ρ ( w T X ) < + ∞ . Then, n E [ − V X ] | β ( V, − w T E [ V X ]) = g ( w ) , V ∈ L q + o ⊆ ∂g ( w ) . Proof. Thanks to Assumption 4.1, the function g defined by (3.3) is convex. Let us define thecontinuous linear operator L : R n → L p by Lw ′ := X T w ′ , w ′ ∈ R n . Then, its adjoint operator L ∗ : L q → R n is given by L ∗ V = E [ V X ] , V ∈ L q . Since g = ρ ◦ L and ρ is finite at w T X , by Z˘alinescu (2002, Theorem 2.8.3(iii)), we have n L ∗ V | V ∈ ∂ρ ( w T X ) o = n E [ V X ] | V ∈ ∂ρ ( w T X ) o ⊆ ∂g ( w ) . (4.7)On the other hand, the subdifferential of the convex lower semicontinuous function ρ at a point Y ∈ L p is given by ∂ρ ( Y ) = (cid:8) − V | β ( V, E [ − V Y ]) = ρ ( Y ) , V ∈ L q + (cid:9) , (4.8)that is, it is the set of maximizers in the dual representation (3.2). Taking Y = w T X in (4.8) andcombining it with (4.7) yields the claim of the lemma.11he next assumption is a constraint qualification for ( D ( r )). Assumption 4.10. There exist V , V ∈ L q ++ such that ˜ α ( V ) ∈ R , α ( V , r ) ∈ R . We are ready to prove the second main theorem of the paper, which establishes the optimalityof a Lagrange multiplier associated to ( D ( r )). Theorem 4.11. Suppose that Assumption 4.1, Assumption 4.2, Assumption 4.6, Assumption 4.10hold, and there exists an optimal solution ( V ∗ , V ∗ , y ∗ ) ∈ L q + × L q + × R for ( D ( r )) . Then, thereexists an optimal Lagrange multiplier w ∗ ∈ R n associated to the inequality constraint of ( D ( r )) .Moreover, every w ∗ ∈ R n that is the Lagrange multiplier of the equality constraint of ( D ( r )) atoptimality is an optimal solution for ( P ( r )) , and ( P ( r )) and ( D ( r )) have the same optimal value p ( r ) . Proof. Let ( V ∗ , V ∗ , y ∗ ) ∈ L q + × L q + × R be an optimal solution for ( D ( r )). Let us denote by d ( r )the optimal value of ( D ( r )). By Assumption 4.10, Slater’s condition holds, that is, there exist V , V ∈ L q ++ , y ∈ R such that ˜ α ( V ) ∈ R , α ( V , r ) ∈ R andmax i ∈{ ,...,n } ( E [ V X i ] + E [ V X i ]) < y since we may simply take y := max i ∈{ ,...,n } ( E [ V X i ] + E [ V X i ]) + 1. Hence, by Borwein, Lewis(1992, Corollary 4.8), there is zero duality gap between ( D ( r )) and its Lagrange dual problem, andwe may write d ( r ) = inf w ∈ R n + sup V ,V ∈ L q + ,y ∈ R (cid:16) ˜ α ( V ) − α ( V , r ) − y − w T ( E [ V X ] + E [ V X ] − y ) (cid:17) = inf w ∈ R n + sup V ,V ∈ L q + ,y ∈ R (cid:16) ˜ α ( V ) − α ( V , r ) − y + E [ − V w T X ] + E [ − V w T X ] + yw T (cid:17) . (4.9)Moreover, Borwein, Lewis (1992, Corollary 4.8) also ensures that there exists an optimal Lagrangemultiplier w ∗ ∈ R n so that d ( r ) = sup V ,V ∈ L q + ,y ∈ R (cid:16) ˜ α ( V ) − α ( V , r ) − y + E [ − V ( w ∗ ) T X ] + E [ − V ( w ∗ ) T X ] + y ( w ∗ ) T (cid:17) , and ( V ∗ , V ∗ , y ∗ ) is an optimal solution of the above concave maximization problem.Let w ∈ R n + . Note that the inner (maximization) problem in (4.9) is easily separated into threeterms assup V ∈ L q + (cid:16) ˜ α ( V ) + E [ − V w T X ] (cid:17) + sup V ∈ L q + (cid:16) − α ( V , r ) + E [ − V w T X ] (cid:17) + sup y ∈ R y (1 − w T ) . (4.10)From the last term in (4.10), it follows immediately thatsup y ∈ R y (1 − w T ) = I W ( w ) . In particular, if w = w ∗ , then we must have w ∗ ∈ W . It is also easy to check that y ∗ ∈ N ( W , w ∗ ) . (4.11)12or the first term in (4.10), note that ρ ( w T X ) = sup V ∈ L q + β ( V , E [ − V w T X ])= sup V ∈ L q + inf n t ∈ R | α − ( V , t ) ≥ E [ − V w T X ] o = sup V ∈ L q + ,x ≥ inf t ∈ R : α − ( V ,t ) ∈ R (cid:16) t + x E [ − V w T X ] − xα − ( V , t ) (cid:17) = sup V ∈ L q + ,x ≥ inf t ∈ R : α − ( V ,t ) ∈ R (cid:16) t + E [ − xV w T X ] − α − ( xV , t ) (cid:17) = sup V ∈ L q + inf t ∈ R : α − ( V ,t ) ∈ R (cid:16) t + E [ − V w T X ] − α − ( V , t ) (cid:17) = sup V ∈ L q + (cid:16) ˜ α ( V ) + E [ − V w T X ] (cid:17) . The steps of this calculation are justified by following the same arguments as in the proof ofProposition 4.7, hence we omit this justification for brevity. In particular, when w = w ∗ , by theoptimality property of V ∗ and property (c) of the definition of maximal risk function, we have g ( w ∗ ) = ρ (( w ∗ ) T X ) = E [ − V ∗ ( w ∗ ) T X ] + ˜ α ( V ∗ ) = β ( V ∗ , E [ − V ∗ ( w ∗ ) T X ]) . By Lemma 4.9, it follows that E [ − V ∗ X ] ∈ ∂g ( w ∗ ) . (4.12)For the second term in (4.10), we first note that V α ( V , r ) is closely related to the supportfunction of the closed convex set A r ; indeed, we have α ( V , r ) = sup Y ∈ A r E [ − V Y ] = σ A r ( − V ) , V ∈ L q + . Hence, by the conjugate duality between indicator function and support function, we havesup V ∈ L q + (cid:16) − α ( V , r ) + E [ − V w T X ] (cid:17) = I A r ( w T X ) = I { g ≤ r } ( w ) . In particular, when w = w ∗ , by the first-order condition, we have − ( w ∗ ) T X ∈ ∂α ( V ∗ , r ) , (4.13)where the subdifferential is with respect to the first variable. Hence, by (4.4), (4.13) is equivalentto − ( w ∗ ) T X ∈ − ∂σ A r ( − V ∗ )as well as to − V ∗ ∈ ∂I A r (( w ∗ ) T X ) . In particular, ∂I A r (( w ∗ ) T X ) = ∅ so that ( w ∗ ) T X ∈ A r , that is, g ( w ∗ ) ≤ r , and − V ∗ ∈ ∂I A r (( w ∗ ) T X ) = N ( A r , ( w ∗ ) T X ) . 13o sup w ∈ R n : g ( w ) ≤ r w T E [ − V ∗ X ] = sup w ∈ R n : g ( w ) ≤ r E [ − V ∗ w T X ] ≤ sup Y ∈ A r E [ − V ∗ Y ] = E [ − V ∗ ( w ∗ ) T X ] , which implies that the inequality in the middle is indeed an equality. Therefore, the equality of thefirst and last quantities yields E [ − V ∗ X ] ∈ N ( { g ≤ r } , w ∗ ) , (4.14)where { g ≤ r } := { w ∈ R n | g ( w ) ≤ r } .Combining the results for the three terms of (4.10) for a generic w ∈ R n + , we see that d ( r ) = inf w ∈ R n + (cid:16) ρ ( w T X ) + I A r ( w T X ) + I W ( w ) (cid:17) = inf n ρ ( w T X ) | ρ ( w T X ) ≤ r, w ∈ W o = p ( r ) , establishing the strong duality between ( P ( r )) and ( D ( r )).Note that the inequality constraint in ( D ( r )) ensures that w T E [ V ∗ X ] ≤ w T ( E [ − V ∗ X ] + y ∗ ) , w ∈ W . (4.15)Moreover, the complementary slackness condition for this constraint yields( w ∗ ) T E [ V ∗ X ] = ( w ∗ ) T ( E [ − V ∗ X ] + y ∗ ) . (4.16)On the other hand, bringing together (4.11) and (4.14) gives E [ − V ∗ X ] + y ∗ ∈ N ( { g ≤ r } , w ∗ ) + N ( W , w ∗ ) = ∂I { g ≤ r } ( w ∗ ) + ∂I W ( w ∗ ) ⊆ ∂ ( I { g ≤ r } + I W )( w ∗ )= ∂I { g ≤ r }∩W ( w ∗ )= N ( { g ≤ r } ∩ W , w ∗ ) . In the above calculation, only the passage to the third line is nontrivial and it is justified by therules of subdifferential calculus; see, for instance, Rockafellar (1970, Theorem 23.8). Hence, by(4.15) and (4.16), we have σ { g ≤ r }∩W ( E [ V ∗ X ]) ≤ σ { g ≤ r }∩W ( E [ − V ∗ X ] + y ∗ ) = ( w ∗ ) T ( E [ − V ∗ X ] + y ∗ ) = ( w ∗ ) T E [ V ∗ X ]Hence, by (4.5), we conclude that E [ V ∗ X ] ∈ N ( { g ≤ r } ∩ W , w ∗ ), that is, E [ − V ∗ X ] ∈ −N ( { g ≤ r } ∩ W , w ∗ ) . Combining this with (4.12), we obtain E [ − V ∗ X ] ∈ ∂g ( w ∗ ) ∩ −N ( { g ≤ r } ∩ W , w ∗ ) . By (4.6), this implies that w ∗ is a minimizer of g over { g ≤ r } ∩ W , that is, w ∗ is an optimalsolution of ( P ( r )). Remark 4.12. It should be noted that we do not work under Assumption 4.3 in Theorem 4.11,hence the strong duality established by Theorem 4.8 is not taken for granted; instead we re-establish strong duality in Theorem 4.11. Then, the reader might naturally question the need forTheorem 4.8. As noted in Remark 4.5, in the absence of Assumption 4.3, the arguments in the proofof Proposition 4.4 (and Theorem 4.8) can be seen as a heuristic way to derive the dual problem( D ( r )). Theorem 4.11 provides the formal justification of this heuristic approach without usingSion’s minimax theorem. 14 Analysis of the problem when the principle risk measure is qua-siconvex In this section, we remove Assumption 4.1 and study ( P ( r )) with ρ being a quasiconvex riskmeasure.Recalling (2.1), we may write p ( r ) = inf n ρ ( w T X ) | ρ ( w T X ) ≤ r, w ∈ W o = inf n t ∈ R | w T X ∈ A t , w T X ∈ A r , w ∈ W o = inf t ∈ R ( t + f ( t, r )) = inf t ∈ R : f ( t,r ) < + ∞ ( t + f ( t, r )) , (5.1)where f ( t, r ) := inf w ∈W (cid:16) I A t ( w T X ) + I A r ( w T X ) (cid:17) , t ∈ R . (5.2)Note that, for each t ∈ R , f ( t, r ) is closely related to the feasibility problem Find w ∈ W such that w T X ∈ A t ∩ A r . ( F P ( t, r ))Indeed, if there exists w ∈ R n solving ( F P ( t, r )), then f ( t, r ) = 0; otherwise, f ( t, r ) = + ∞ . Theexpression in (5.2) formulates ( F P ( t, r )) as an optimization problem whose optimal value is f ( t, r );this problem is convex because A t , A r are convex sets. Hence, in view of (5.1), the quasiconvexportfolio optimization problem ( P ( r )) is characterized by a family of convex optimization problems;this is a well-known paradigm in quasiconvex programming as discussed recently in Agrawal, Boyd(2020, Section 2.1).We introduce an analogue of Assumption 4.10 that will be needed in recovering a solution for( F P ( t, r )) below. Assumption 5.1. Given t ∈ R , there exist V , V ∈ L q ++ such that α ( V , t ) ∈ R , α ( V , r ) ∈ R . In the next theorem, following a similar path as in Section 4 (see Theorem 4.8, Theorem 4.11),we provide a dual formulation of f ( t, r ) and a method to calculate a solution for ( F P ( t, r )). Theorem 5.2. Let t ∈ R and consider the problemmaximize − α ( V , t ) − α ( V , r ) − y ( F D ( t, r )) subject to E [ V X ] + E [ V X ] ≤ y V , V ∈ L q + , y ∈ R , (i) Then, ( F P ( t, r )) and ( F D ( t, r )) have the same optimal value f ( t, r ) .(ii) Suppose that Assumption 5.1 holds for t and there exists an optimal solution ( V t , V t , y t ) for ( F D ( t, r )) . Then, there exists an optimal Lagrange multiplier w t ∈ R n associated to theinequality constraint of ( F D ( t, r )) . Moreover, every w t ∈ R n that is the Lagrange multiplierof the inequality constraint of ( F D ( t, r )) at optimality is an optimal solution for ( F P ( t, r )) . Proof. We first prove (i) under the additional assumption that α ( V , t ) ∈ R , α ( V , r ) for all V , V ∈ L q + . Since A t , A r are closed convex subsets of L p , similar to the proof of Proposition 4.4,15e have f ( t, r ) = inf w ∈W (cid:16) I A t ( w T X ) + I A r ( w T X ) (cid:17) = inf w ∈W sup V ∈ L q + (cid:16) E [ − V w T X ] − α ( V , t ) (cid:17) + sup V ∈ L q + (cid:16) E [ − V w T X ] − α ( V , r ) (cid:17)! = inf w ∈W sup V ∈ L q + ,V ∈ L q + v t,r ( w, V , V ) , where v t,r ( w, V , V ) := E [ − V w T X ] + E [ − V w T X ] − α ( V , t ) − α ( V , r ) . By the properties of support function, it is clear that ( V , V ) v t,r ( w, V , V ) is concave and uppersemicontinuous for fixed w ∈ W . On the other hand, for fixed ( V , V ) ∈ L q + × L q + , the function w v t,r ( w, V , V ) is continuous and affine, hence lower semicontinuous convex. Since W is aconvex compact set and v t,r has finite values thanks to our additional assumption, we may applythe standard minimax theorem Sion (1958, Corollary 3.3) and get f ( t, r ) = sup V ∈ L q + ,V ∈ L q + inf w ∈W v t,r ( w, V , V )= sup V ∈ L q + ,V ∈ L q + (cid:16) inf n ( E [ − V X ] + E [ − V X ]) T w | T w = 1 , w ∈ R n + o − α ( V , t ) − α ( V , r ) (cid:17) . Note that the inner minimization problem is a finite-dimensional linear optimization problem withnonempty feasible region. Hence, by linear programming duality, we may pass to its dual formula-tion, which yields f ( t, r ) = sup V ∈ L q + ,V ∈ L q + (sup {− y | E [ V X ] + E [ V X ] ≤ y , y ∈ R } − α ( V , t ) − α ( V , r ))= sup (cid:8) − α ( V , t ) − α ( V , r ) − y | E [ V X ] + E [ V X ] ≤ y , V , V ∈ L q + , y ∈ R (cid:9) , which coincides with the optimal value of ( F D ( t, r )).Next, we prove (i) without the additional assumption as well as (ii). Let ˜ f ( t, r ) be the optimalvalue of ( F D ( t, r )). Let ( V t , V t , y t ) ∈ L q + × L q + × R be an optimal solution for ( F D ( t, r )). ByAssumption 5.1, there exist V , V ∈ L q ++ such that α V , t ) ∈ R and α ( V , r ) ∈ R . Similar tothe proof of Theorem 4.11, it follows that Slater’s condition holds for ( F D ( t, r )); hence, by strongduality for convex optimization, there exists an optimal Lagrange multiplier w t ∈ R n such that˜ f ( t, r )= inf w ∈ R n sup V ,V ∈ L q + ,y ∈ R (cid:16) − α ( V , t ) − α ( V , r ) − y − w T ( E [ V X ] + E [ V X ] − y ) (cid:17) = inf w ∈ R n sup V ,V ∈ L q + ,y ∈ R (cid:16) − α ( V , t ) − α ( V , r ) − y − E [ V w T X ] + E [ V w T X ] − yw T (cid:17) = sup V ,V ∈ L q + ,y ∈ R (cid:16) − α ( V , t ) − α ( V , r ) − y − E [ V ( w t ) T X ] − E [ V ( w t ) T X ] − y ( w t ) T (cid:17) = sup V ∈ L q + (cid:16) − α ( V , t ) − E [ V ( w t ) T X ] (cid:17) + sup V ∈ L q + (cid:16) − α ( V , r ) − E [ V ( w t ) T X ] (cid:17) + sup t ∈ R y (cid:16) − ( w t ) T (cid:17) . f ( t, r ) = f ( t, r ) so that (i) holds without the additional assumption as well. Moreover, it can be checkedthat w t ∈ W , and y t ∈ N ( W , w t ) , − ( w t ) T X ∈ ∂α ( V t , t ) , ( w t ) T X ∈ ∂α ( V t , r ) , which implies that E [ − V t X ] ∈ −N ( { g ≤ t } , w t ) , E [ − V t X ] ∈ −N ( { g ≤ r } , w t );and finally we obtain E [ − V t X ] ∈ −N ( { g ≤ t } ∩ { g ≤ r } ∩ W , w t ) . Hence, we conclude that w t solves the feasibility problem ( F P ( t, r )).The next assumption will be useful when devising a method to find an approximately optimalsolution for ( P ( r )). Assumption 5.3. It holds g ( ) ∈ R and dom g ∩{ g ≤ r }∩W 6 = ∅ . In other words, ρ ( T X ) ∈ R and there exists w ∈ W such that ρ (( w ) T X ) ∈ R and ρ (( w ) T X ) ≤ r . Finally, we discuss a simple method to solve ( P ( r )) with the help of Theorem 5.2. First notethat p ( r ) = inf { g ( w ) | g ( w ) ≤ r, w ∈ W} . Since { g ≤ r }∩W is a convex set and g is a lower semicontinuous function, ( P ( r )) has an optimalsolution, that is, there exists w ∗ ∈ W such that g ( w ∗ ) ≤ r and g ( w ∗ ) = p ( r ) . Moreover, under Assumption 5.3, we also have p ( r ) = g ( w ∗ ) ≤ g ( w ) < + ∞ . On the other hand, by the monotonicity of ρ and Assumption 5.3, −∞ < g ( ) = ρ ( T X ) ≤ ρ (( w ∗ ) T X ) = g ( w ∗ ) = p ( r )Hence, p ( r ) ∈ R with finite upper bound u := g ( w ) and finite lower bound ℓ := g ( ). Let ε > P ( r )) as follows. At each iteration k ∈ N , we start with ℓ k , u k ∈ R such that ℓ k ≤ p ( r ) ≤ u k and we let t k := ℓ k + u k . Then, under Assumption 4.10, we solve the feasibility problem ( F P ( t k , r )), that is, we calculate f ( t k , r ). If f ( t k , r ) = 0, then we have ℓ k ≤ p ( r ) ≤ t k , in which case we proceed to the next iterationusing ℓ k +1 := ℓ k and u k +1 := t k . Otherwise, if f ( t k , r ) = + ∞ , then we have t k ≤ p ( r ) ≤ u k ,in which case we proceed to the next iteration using ℓ k +1 := t k and u k +1 := u k . We stop thisprocedure at the first iteration number K for which u K − ℓ K ≤ ε . It can be checked that K ≤ (cid:24) log (cid:18) g ( w ) − g ( ) ε (cid:19)(cid:25) so that the algorithm stops in finitely many iterations. Then, we may apply Theorem 5.2 and findan optimal Lagrange multiplier w t K , which also solves ( F P ( t K , r )). Hence, w t K ∈ W , g ( w t K ) ≤ r and p ( r ) ≤ g ( w t K ) ≤ t K ≤ p ( r ) + ε, which shows that w t K is an ε -optimal solution for ( P ( r )).17 Examples In this section, we consider some well-known classes of quasiconvex risk measures as special casesof ρ and ρ . Let ρ be a lower semicontinuous convex risk measure on L p with ρ (0) ∈ R and acceptance sets( A t ) t ∈ R . Since ρ is translative, its acceptance set A := A at level 0 determines ρ completely. Asa result, the dual representation in Proposition 2.2 reduces to a simpler form which we derive herefor the convenience of the reader. Let V ∈ L q + \{ } and t ∈ R . Then, by the translativity of ρ , α ( V, t ) = sup Y ∈ L p : ρ ( Y ) ≤ t E [ − V Y ] = sup Y ∈ L p : ρ ( Y + t ) ≤ E [ − V Y ] = sup Y ∈ L p : Y + t ∈ A E [ − V Y ] = γ ( V ) + t E [ V ] , where γ ( V ) := sup Y ∈ A E [ − V Y ] . The function γ : L q + \{ } → ¯ R is called the minimal penalty function of ρ in the sense of convexrisk measures (not to be confused with the minimal penalty function α in the sense of quasiconvexrisk measures ); it follows from the definition that γ is convex and lower semicontinuous. Since V = 0, we have E [ V ] > 0. From this and the above expression for α , it is evident that t α ( V, t )is concave (indeed affine) and continuous withlim t →∞ α − ( V, t ) = lim t →∞ α ( V, t ) = + ∞ (see Assumption 4.6); and t α ( V, t ) has a true inverse given by β ( V, s ) = s − γ ( V ) E [ V ] , s ∈ R . Moreover, for each a ∈ R , the set n ( V, s ) ∈ L q, × R | β ( V, s ) ≥ a o = n ( V, s ) ∈ L q, × R | γ ( V ) + a E [ V ] − s ≤ o is closed by the lower semicontinuity of γ ; therefore, β is jointly upper semicontinuous on L q, × R (see Assumption 4.2). On the other hand, we have˜ α ( V ) := inf t ∈ R ( t − α ( V, t )) = inf t ∈ R ((1 − E [ V ]) t − γ ( V )) = ( − γ ( V ) if E [ V ] = 1 , −∞ else . Now suppose that, in ( P ( r )), both ρ and ρ are convex risk measures with respective minimalpenalty functions γ and γ in the sense of convex risk measures. Then, by the above discussion,Assumption 4.1, Assumption 4.2 and Assumption 4.6 hold, and the dual problem ( D ( r )) can berewritten as maximize − γ ( V ) − γ ( V ) − r E [ V ] − y subject to E [ V X ] + E [ V X ] ≤ y E [ V ] = 1 V , V ∈ L q + , y ∈ R . xample 6.1. For this example, we assume that p = + ∞ (and q = 1). For each j ∈ { , } , let ussuppose that ρ j is the entropic risk measure with risk aversion parameter r j > ρ j ( Y ) = 1 r j log E (cid:2) e − r j Y (cid:3) , Y ∈ L ∞ . In this case, it is well-known that γ j is the relative entropy function given by γ j ( V ) = 1 r j E (cid:20) V log (cid:18) V E [ V ] (cid:19)(cid:21) = 1 r j ( E [ V log( V )] − E [ V ] log( E [ V ])) , V ∈ L \{ } . Note that Assumption 4.10 is satisfied here: by taking V = V ≡ 1, we have ˜ α ( V ) = 0 ∈ R and α ( V , r ) = r ∈ R . Moreover, ( D ( r )) takes the formmaximize − r E [ V log( V )] − r E [ V log( V )] + 1 r E [ V ] log( E [ V ]) − r E [ V ] − y subject to E [ V X ] + E [ V X ] ≤ y E [ V ] = 1 V , V ∈ L , y ∈ R . In the case of a finite probability space, this problem can be solved numerically using, for instance,the convex optimization package CVX (see Grant, Boyd (2020, 2008)) as it is able to work withthe convex function z z log( z ) on R + . As is standard in convex optimization, these packagesalso provide the value of the Lagrange multiplier w ∗ that corresponds to the inequality constraintin ( D ( r )) at (approximate) optimality. By Theorem 4.11, such w ∗ is an (approximately) optimalsolution of ( P ( r )). Certainty equivalents form an important class of quasiconvex risk measures. We briefly recall theirdefinitions and properties; see Drapeau, Kupper (2013, Example 8) for more details. To avoidintegrability issues, we assume that p = + ∞ (hence q = 1) in all examples although, in eachexample, a larger L p space can be considered depending on the nature of the loss function.Let ℓ : R → ( −∞ , + ∞ ] be a convex lower semicontinuous increasing function that is differen-tiable on dom ℓ := { y ∈ R | ℓ ( y ) < + ∞} , we call ℓ a loss function . Let ρ be the certainty equivalent corresponding to ℓ , that is, ρ ( Y ) = ℓ − ( E [ ℓ ( − Y )]) , Y ∈ L ∞ , where ℓ − is the left-continuous inverse of ℓ . The minimal penalty function of ρ is given by α ( V, t ) = E (cid:20) V h (cid:18) λ ( V, t ) V E [ V ] (cid:19)(cid:21) , V ∈ L \{ } , t ∈ R , (6.1)where h is the right-continuous inverse of the derivative ℓ ′ of ℓ , and λ ( V, t ) > E (cid:20) ℓ (cid:18) h (cid:18) λ ( V, m ) V E [ V ] (cid:19)(cid:19)(cid:21) = ℓ + ( t ) , (6.2)where ℓ + is the right-continuous version of ℓ .Next, we consider some special cases of ℓ for which more explicit forms of α can be obtained.19 xample 6.2. Suppose that ℓ is the quadratic loss function given by ℓ ( y ) = ( y + y if y ≥ − , − else . Let V ∈ L \{ } , t, s ∈ R . After elementary calculations, we may solve (6.2) for λ ( V, t ) and usethe resulting expression in (6.1) to get α ( V, t ) = ( (1 + t ) k V k − E [ V ] if t > − , − E [ V ] else , β ( V, s ) = ( s + E [ V ] k V k − s > − E [ V ] , −∞ else . In particular, as in Section 6.1, it is easy to verify that Assumption 4.2 and Assumption 4.6 holdfor α = α . Example 6.3. Suppose that ℓ is the logarithmic loss function given by ℓ ( y ) = ( − log( − y ) if y < , + ∞ else . Let V ∈ L \{ } , t, s ∈ R . Then, after some elementary calculations, we obtain α ( V, t ) = ( te E [log( V )] if t < , + ∞ else , β ( V, s ) = ( se − E [log( V )] if s < , . It follows that Assumption 4.2 and Assumption 4.6 hold for α = α .We conclude the paper with an example where ρ , ρ are assumed to be certainty equivalentswhose respective loss functions ℓ , ℓ are among the two examples described above, which illustratesa possible special form of ( F D ( t, r )). Example 6.4. Suppose that ℓ is the quadratic loss function in Example 6.2 and ℓ is the logarith-mic loss function in Example 6.3. Assume that r < t ∈ R . If t ≤ − 1, then ( F D ( t, r ))can be rewritten as maximize − E [ V ] − re E [log( V )] − y subject to E [ V X ] + E [ V X ] ≤ y V , V ∈ L , y ∈ R . 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