Optimality Conditions for Nonsmooth Generalized Semi-Infinite Programs

Abstract

We consider a class of nonsmooth generalized semi-infinite programming problems. We apply results from parametric optimization to the lower level problems of generalized semi-infinite programming problems to get estimates for the value functions of the lower level problems and thus derive necessary optimality conditions for generalized semi-infinite programming problems. We also derive some new estimates for the value functions of the lower level problems in terms of generalized differentiation and further obtain the necessary optimality conditions.

1. Introduction

Generalized semi-infinite programming problem (GSIP) is of the form min𝑓(𝑥)s.t.𝑥∈ℝ^𝑛,𝑔𝑥,𝑦≤0,𝑦∈𝑌(𝑥), (1) where 𝑌(𝑥)∶={𝑦∈ℝ^𝑚∣𝑣(𝑥,𝑦)≤0}. GSIP is different from the standard semi-infinite programming in that its index set 𝑌 is dependent on 𝑥.

This first systematic study of GSIP was Hettich and Still [1] where the reduction method was used to reduce GSIP into standard nonlinear programming problems and second-order optimality conditions were derived. Necessary optimality conditions at an optimal solution 𝑥 for (1) with differentiable data are as follows there exist nonnegative numbers 𝜆₀,…,𝜆_𝑝, not all zero, such that 𝜆₀∇𝑓(𝑥)+^𝑝_𝑗=1𝜆_𝑗∇_𝑥𝐿𝑥,𝑦^𝑗,𝛼^𝑗,𝛽^𝑗=0, (2) where 𝐿(𝑥,𝑦,𝛼,𝛽)=𝛼𝑔(𝑥,𝑦)−⟨𝛽,𝑣(𝑥,𝑦)⟩ and each (𝛼^𝑗,𝛽^𝑗) is the usual FJ-multiplier of the lower level problem at its optimal solution 𝑦^𝑗𝑄(𝑥)max𝑔𝑥,𝑦s.t.𝑣𝑥,𝑦≤0. (3) This condition was first derived by Jongen et al. [2] in an elementary way without any constraint qualifications or any kind of reduction approaches. They also proposed a constraint qualification under which it follows that 𝜆₀>0 and discussed some geometrical properties of the feasible set which do not appear in standard semi-infinite case. The optimality conditions are further explored by Rückmann and Shapiro [3] and Stein [4].

GSIP in itself is of complex and exclusive structures such as the nonclosedness of the feasible set, nonconvexity, nonsmoothness, and bilevel structure and thus a difficult problem to solve see, for example, [2, 5, 6]. We also refer to [7–10] for some recent study on the structure of GSIP.

It is obvious that GSIP can be rewritten equivalently as the nonlinear programming problem (𝑥)(𝑥)min𝑓s.t.𝜙≤0, (4) where 𝜙(𝑥) is the optimal value of 𝑄(𝑥). Then we can relate GSIP to the min-max problem min_𝑥max𝑓(𝑥)−𝑓(𝑥),𝜙(𝑥); (5) see [3] for more details. On the other hand, GSIP can be related to the following bilevel problem: min_𝑥,𝑦𝑓(𝑥)𝑔(𝑥.s.t.𝑔𝑥,𝑦≤0,𝑦∈argmax𝑥,𝑦∶𝑦∈𝑌) (6) The problem (6) is a special bilevel optimization problem in that its upper level constraint is the same as the objective function of its lower level problem. However, there is a slight difference between GSIP problem (1) and problem (6). The feasible set of (6) is a subset of (1) in that the feasible set of (1) is the combination of the feasible set of (6) and the complement of dom𝑌. For more comparisons between GSIP and bilevel problems, see Stein and Still [11].

The bilevel problem (6) is equivalent to the following problem: min_𝑥,𝑦𝑓(𝑥)s.t.𝑥,𝑦∈Ω,𝐺𝑥,𝑦≤0, (7) where Ω={(𝑥,𝑦)∣𝑔(𝑥,𝑦)≤0,𝑣(𝑥,𝑦)≤0} and 𝐺(𝑥,𝑦)=𝜙(𝑥)−𝑔(𝑥,𝑦). This approach was used by Dempe and Zemkoho [12] to study bilevel optimization problems. For general references of bilevel optimization, see [13].

In this paper we concentrate on the optimality conditions of nonsmooth GSIP whose defining functions are Lipschitz continuous. Similar works are [14, 15]. We achieve this via the lower level optimal value function reformulation and then derive its necessary optimality conditions via the generalized differentiation. One of the key steps is to estimate the generalized gradients of the lower level optimal value function which involves parametric optimization. We will consider two cases with different approaches related to the two reformulations of GSIP as previously mentioned. Firstly, we develop optimality conditions via the min-max formulation with Lipschitz lower level optimal value function. Secondly, we develop optimality conditions via bilevel formulation under the assumption of partial calmness.

2. Preliminaries

In this section, we present some basic definitions and results from variational analysis [16, 17]. Given a set 𝐴 in ℝ^𝑛, the regular normal cone 𝑁_𝐴 of 𝐴 at 𝑥∈𝐴 is defined by 𝑁_𝐴(𝑥)=𝑣∈ℝ^𝑛∣limsup_𝑥𝐴−→_𝑥⟨𝑣,𝑥−𝑥⟩‖𝑥−𝑥‖≤0. (8) The (general) normal cone 𝑁_𝐴 of 𝐴 at 𝑥 is defined by 𝑁_𝐴(𝑥)=limsup_𝑥→𝑥𝑁_𝐴(𝑥)=𝑣∣∃𝑥^𝑘⟶𝑥,∃𝑣^𝑘⟶𝑣with𝑣^𝑘∈𝑁_𝐴𝑥^𝑘. (9) Given a function 𝑓∶ℝ^𝑛→ℝ and a point 𝑥 with 𝑓(𝑥) finite, denote by epi𝑓 the epigraph of 𝑓. The regular subdifferential of 𝑓 at 𝑥 is defined by 𝜕𝑓(𝑥)∶=𝑢∈ℝ^𝑛∣liminf_{𝑥→𝑥,𝑥≠𝑥}𝑓(𝑥)−𝑓(𝑥)−⟨𝑢,𝑥−𝑥⟩‖𝑥−𝑥‖≥0. (10) The general (basic, limiting) and singular subdifferential of 𝑓 at 𝑥 are defined, respectively, by 𝜕𝑓(𝑥)∶=limsup_𝑥𝑓−→_𝑥(𝑥),𝜕𝜕𝑓_∞𝑓(𝑥)∶=limsup_𝑥𝑓−→_{𝑥,𝜆↓0}𝜆𝜕𝑓(𝑥). (11) The upper regular subdifferential of 𝑓 at 𝑥 is defined by 𝜕⁺𝑓(𝑥)∶=−𝜕(−𝑓)(𝑥), and the upper subdifferential of 𝑓 is defined by 𝑥𝜕⁺𝑓(𝑥)∶=limsup_𝑥𝑓−→_𝑥𝜕⁺𝑓(𝑥).

The Clarke (convexified) normal cone can be defined by two different approaches. On the one hand, it can be defined by the polar cone of the Clarke’s tangent cone 𝑁_𝐴(𝑥)∶=𝑇_𝐴(𝑥)^∘, (12) where 𝑇_𝐴(𝑥)=liminf_𝑦𝐴−→_{𝑥,𝑡↘0}(𝐴−𝑦)/𝑡 or defined via the generalized directional derivative of the (Lipschitzian) distant function dist(⋅,𝐴); see Clarke [18]. On the other hand, it can be defined by the closed convex hull of the (general) normal cone 𝑁_𝐴(𝑥)∶=clco𝑁_𝐴(𝑥). (13) For this definition and also the equivalence of the two definitions, see, for example, Rockafellar and Wets [16].

The Clarke subgradients and Clarke horizon subgradients of 𝑓 at 𝑥 are defined by 𝜕𝑓(𝑥)∶=𝑣∣(𝑣,−1)∈𝑁_epi𝑓𝑥,𝑓(𝑥),𝜕^∞𝑓(𝑥)∶=𝑣∣(𝑣,0)∈𝑁_epi𝑓𝑥,𝑓(𝑥). (14) The relationship between the Clarke sub-subdifferentials and basic sub-differentials is also referred to by Mordukhovich [17, Theorem 3.57].

Proposition 1 (see [16]). Let 𝑓 be proper and lsc around 𝑥∈dom𝑓. Then, 𝜕𝑓(𝑥)=clco𝜕𝑓(𝑥)+𝜕^∞𝑓(𝑥). (15) If, in particular, 𝑓 is Lipschitz continuous at 𝑥, then 𝜕𝑓(𝑥)=clco𝜕𝑓(𝑥). (16)

The normal cone 𝑁_𝐴 enjoys the robustness property 𝑁_𝐴(𝑥)=limsup_𝑥→𝑥𝑁_𝐴(𝑥) provided that the setting is finite dimension [17, page 11]. However, this is not true for the convexified cone 𝑁_𝐴; see, for example, Rockafellar [19]. Consider 𝐴∶=𝑥∈ℝ³∣𝑥₃=𝑥₁𝑥₂or𝑥₃=−𝑥₁𝑥₂,𝑥=(0,0,0). (17)(18) The normal cone 𝑁_𝐴(𝑥) is just the 𝑥₃-axis, but 𝑁_𝐴(𝑥) is the 𝑥₂𝑥₃-plane for all 𝑥=(𝑥₁,0,0). The following proposition is from Rockafellar [19].

Proposition 2 (see [19]). If 𝐴 is convex, or if 𝑁_𝐴(𝑥) is pointed, then the multifunction 𝑁_𝐴 is closed at 𝑥; that is, for all 𝑥^𝑘→𝑥, 𝑦−−−−−−→^{𝑘𝑁𝐴(𝑥𝑘)}𝑦, one has 𝑦∈𝑁_𝐴(𝑥).

Proposition 3. The Clarke normal cone has the robustness property 𝑁_𝐴(𝑥)=limsup_𝑦→𝑥𝑁_𝐴𝑦 (19) provided that 𝑁_𝐴(𝑥) is pointed.

Proof. It suffices to prove that limsup_𝑦→𝑥co𝑁_𝐴(𝑦)⊂clcolimsup_𝑦→𝑥𝑁_𝐴(𝑦)=𝑁_𝐴(𝑥). Let 𝑣∈limsup_𝑦→𝑥co𝑁_𝐴(𝑦). Then there are 𝑦_𝑘∈𝐴, 𝑣_𝑖𝑘∈𝑁_𝐴(𝑦_𝑘), 𝑖=1,…,𝑛+1 such that ^𝑛+1_𝑖=1𝑣_𝑖𝑘⟶𝑣as𝑘⟶∞, (20) since the sets 𝑁_𝐴(𝑦_𝑘) are cones. Let 𝜆_𝑘=∑^𝑛+1_𝑖=1‖𝑣_𝑖𝑘‖. Then {𝜆_𝑘} is bounded; that is also to say, {𝑣_𝑖𝑘} are bounded for all 𝑖. Otherwise, ∑^𝑛+1_𝑖=1𝑣_𝑖𝑘/𝜆_𝑘→0. That is, 𝑣₁+⋯+𝑣_𝑛+1=0, where 𝑣_𝑖 is the limit of {𝑣_𝑖𝑘/𝜆_𝑘}_𝑘 for each 𝑖. Note that 𝑣_𝑖∈𝑁_𝐴(𝑥) since 𝑁_𝐴(𝑥)=limsup_𝑦→𝑥𝑁_𝐴(𝑦). Thus 𝑣₁=⋯=𝑣_𝑛+1=0 by the pointedness of 𝑁_𝐴(𝑥). On the other hand, ∑_𝑖‖𝑣_𝑖‖=1. This is a contradiction. Thus the sequence {𝑣_𝑖𝑘} is bounded. By taking subsequences, we may assume that 𝑣_𝑖𝑘→𝑣_𝑖. Then 𝑣_𝑖∈𝑁_𝐴(𝑥) and 𝑣=𝑣₁+⋯+𝑣_𝑛+1. This completes the proof.

The following definitions are required for further development.

Definition 4 (see [17, Definition  1.63]). Let 𝑆∶𝑋⇉𝑌 be a set-valued mapping with 𝑥∈dom𝑆, the domain of 𝑆. (i) Given 𝑦∈𝑆(𝑥), we say that the mapping 𝑆 is inner semicontinuous at (𝑥,𝑦) if for every sequence 𝑥_𝑘→𝑥 there is a sequence 𝑦_𝑘∈𝑆(𝑥_𝑘) converging to 𝑦 as 𝑘→∞. (ii)𝑆 is inner semicompact at 𝑥 if for every sequence 𝑥_𝑘→𝑥 there is a sequence 𝑦_𝑘∈𝑆(𝑥_𝑘) that contains a convergent subsequence as 𝑘→∞.(iii)𝑆 is 𝜇-inner semicontinuous at (𝑥,𝑦) (𝜇-inner semicompact at 𝑥) if in above two cases, 𝑥_𝑘→𝑥 is replaced by 𝑥_𝑘→𝑥 with 𝜇(𝑥_𝑘)→𝜇(𝑥).

Here the concept of 𝜇-inner semicontinuity/semicompactness is important for our considerations. It is typical that the value function 𝜙 of the lower level problem 𝑄(𝑥) of GSIP is not continuous, even taking value −∞.

Theorem 5 (subdifferentiation of maximum functions [17, Theorem  3.46]). Consider the maximum function of the form max𝜙_𝑖(𝑥)=max𝜙_𝑖(𝑥)∣𝑖=1,…,𝑙. (21) Let 𝜙_𝑖 be lower semicontinuous around 𝑥 for 𝑖∈𝐼(𝑥)∶={𝑖∣max(𝜙_𝑖)(𝑥)=𝜙_𝑖(𝑥)} and be upper semicontinuous at 𝑥 for 𝑖∉𝐼(𝑥). Assume that the qualification holds: _{(𝑖∈𝐼𝑥)}𝑥^∗_𝑖=0,𝑥^∗_𝑖∈𝜕^∞𝜙_𝑖(𝑥)⟹𝑥^∗_𝑖=0,𝑖∈𝐼(𝑥). (22) Then 𝜕max𝜙_𝑖(𝑥)⊂_{(𝑖∈𝐼𝑥)}𝜆_𝑖𝜕𝜙_𝑖(𝑥)∣𝜆₁,…,𝜆_𝑙∈Λ(𝑥), (23) where Λ(𝑥)={(𝜆₁,…,𝜆_𝑙)∣𝜆_𝑖≥0,∑^𝑙_𝑖=1𝜆_𝑖=1,𝜆_𝑖(𝜙_𝑖(𝑥)−(max𝜙_𝑖)(𝑥))=0}.

Note that qualification (22) always holds if all related functions are locally Lipschitz.

The following two results are about continuity properties and estimates of subdifferentials of marginal functions which are crucial to our analysis for GSIP problems.

Proposition 6 (limiting subgradients of marginal functions [20]). Consider the parametric optimization problem 𝜇(𝑥)∶=inf𝜑𝑥,𝑦∣𝑦∈𝐺(𝑥), (24) and let 𝑀(𝑥)∶={𝑦∈𝐺(𝑥)∣𝜇(𝑥)=𝜑(𝑥,𝑦)}, 𝐺(𝑥)∶={𝑦∈ℝ^𝑚∣𝜑_𝑖(𝑥,𝑦)≤0,𝑖=1,…,𝑙}. For simplicity, one does not consider the case with equality constraints involved. (i)Assume that 𝑀 is 𝜇-inner semicontinuous at (𝑥,𝑦)∈gph𝑀 (the graph of 𝑀), that 𝜑 and all 𝜑_𝑖 are Lipschitz continuous around (𝑥,𝑦), and that the following qualification condition is satisfied:𝑜𝑛𝑙𝑦𝑡ℎ𝑒𝑣𝑒𝑐𝑡𝑜𝑟𝜆₁,…,𝜆_l=0∈ℝ^l𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠𝑡ℎ𝑒𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛0∈^𝑙_𝑖=1𝜆_𝑖𝜕𝜑_𝑖𝑥,𝑦𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝜆₁,…,𝜆_𝑙∈ℝ^𝑙₊𝑤𝑖𝑡ℎ𝜆_𝑖𝜑_𝑖𝑥,𝑦=0,𝑖=1,…,𝑙. (25)
One has the inclusions 𝜕𝜇(𝑥)⊂𝑢^∗∈𝑋^∗∣𝑢^∗,0∈𝜕𝜑𝑥,𝑦+^𝑙_𝑖=1𝜆_𝑖𝜕𝜑_𝑖𝑥,𝑦𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝜆₁,…,𝜆_𝑙∈ℝ^𝑙₊𝑤𝑖𝑡ℎ𝜆_𝑖𝜑_𝑖𝑥,𝑦=0,𝑖=1,…,𝑙. (26)(ii)Assume that 𝑀 is 𝜇-inner semicompact at 𝑥, and all 𝜑 and 𝜑_𝑖 are Lipschitz continuous around (𝑥,𝑦) for all 𝑦∈𝑀(𝑥), and qualification (25) holds for all (𝑥,𝑦), 𝑦∈𝑀(𝑥). Then 𝜕𝜇(𝑥)⊂_{(𝑦∈𝑀𝑥)}𝑢^∗∈𝑋^∗∣𝑢^∗,0∈𝜕𝜑𝑥,𝑦+^𝑙_𝑖=1𝜆_𝑖𝜕𝜑_𝑖𝑥,𝑦,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝜆₁,…,𝜆_𝑙∈ℝ^𝑚₊𝑤𝑖𝑡ℎ𝜆_𝑖𝜑_𝑖𝑥,𝑦=0,𝑖=1,…,𝑙. (27)

Proposition 7 (Lipschitz continuity of marginal functions [21, Theorem 5.2]). Continue to consider the parametric problem (24) in Proposition 6. Then the following assertions hold. (i)Assume that 𝑀 is 𝜇-inner semicontinuous at (𝑥,𝑦)∈gph𝑀 and 𝜑 is locally Lipschitz around this point. Then 𝜇 is Lipschitz around 𝑥 provided that it is lsc around 𝑥 and 𝐺 is Lipschitz-like around (𝑥,𝑦).(ii)Assume that 𝑀 is 𝜇-inner compact at 𝑥 and 𝜑 is locally Lipschitz around (𝑥,𝑦) for all 𝑦∈𝑀(𝑥). Then 𝜇 is Lipschitz around 𝑥 provided that it is lsc around 𝑥 and 𝐺 is Lipschitz-like around (𝑥,𝑦) for all 𝑦∈𝑀(𝑥).

3. Main Results

Now we are prepared to develop the optimality conditions for GSIP problem (1). Given a local solution 𝑥 of problem (1), associate it with the following min-max problem min_𝑥max𝑓(𝑥)−𝑓(𝑥),𝜙(𝑥), (28) where 𝜙(𝑥)∶=sup_{𝑦∈𝑌(𝑥)}𝑔(𝑥,𝑦). Let 𝑌₀(𝑥)∶={𝑦∈𝑌(𝑥)∶𝜙(𝑥)=𝑔(𝑥,𝑦)}. Denote by 𝐹(𝑥)=max𝑓(𝑥)−𝑓(𝑥),𝜙(𝑥). (29) If 𝑥 solves GSIP (1), then 𝑥 also solves problem min𝐹(𝑥)∶𝑥∈ℝ^𝑛, (30) and thus by generalized Fermat's rule (cf. [16, Theorem 10.1]), we have 0∈𝜕𝐹(𝑥). (31) So, calculus for the maximum function and the estimate of subdifferentials are essential to proceed. From (31) and Theorem 5, there exists 𝜇∈[0,1] such that (if 𝜙 is Lipschitz) 0∈𝜇𝜕𝑓(𝑥)+1−𝜇𝜕𝜙(𝑥)⊂𝜇𝜕𝑓(𝑥)+1−𝜇co𝜕𝜙(𝑥). (32) Note that for a Lipschitz function 𝜙, co𝜕𝜙(𝑥)=−co𝜕−𝜙(𝑥). (33)

Theorem 8 (optimality for GSIP with Lipschitz lower level optimal value function). Consider the GSIP problem (1), and let 𝑥∈𝑀 be its locally optimal solution. Assume that all functions 𝑓, 𝑔, and 𝑣 are Lipschitz continuous, 𝑌₀ is 𝜙-inner semicompact at 𝑥, and 𝑌 is Lipschitz-like at (𝑥,𝑦) for all 𝑦∈𝑌₀(𝑥). Then there is 𝑦^𝑗∈𝑌₀(𝑥) and 𝜆₀≥0,  𝜆_𝑗≥0,  𝛼^𝑗_𝑖≥0, 𝑖=1,…,𝑙, 𝑗=1,…,𝑘 such that ∑^𝑘_𝑗=1𝜆_𝑗=1 and 0∈𝜆₀𝜕𝑓(𝑥),0+^𝑘_𝑗=1𝜆_𝑗−𝜕−𝑔𝑥,𝑦^𝑗−^𝑙_𝑖=1𝛼^𝑗_𝑖𝜕𝑣_𝑖𝑥,𝑦^𝑗. (34) If in addition 𝑔 and all components of 𝑣 are regular at all (𝑥,𝑦^𝑗), then the optimality is of the form 0∈𝜆₀𝜕𝑓(𝑥)+^𝑘_𝑗=1𝜆_𝑗−𝜕_𝑥−𝑔𝑥,𝑦^𝑗−^𝑙_𝑖=1𝛼^𝑗_𝑖𝜕_𝑥𝑣_𝑖𝑥,𝑦^𝑗,0∈^𝑘_𝑗=1𝜆_𝑗−𝜕_𝑦−𝑔𝑥,𝑦^𝑗−^𝑙_𝑖=1𝛼^𝑗_𝑖𝜕_𝑦𝑣_𝑖𝑥,𝑦^𝑗. (35) Note that −𝜕(−𝑔)=𝜕⁺𝑔.

Proof. Under regularity and Lipschitz continuity, since the following calculus rule for basic subgradients holds (see [16, Corollary  10.11]): 𝜕_𝑥𝑓𝑥,𝑦=𝑢∣∃𝑣s.t.(𝑢,𝑣)∈𝜕𝑓𝑥,𝑦, (36) the last two equations follow directly from the first one. Note that the function −𝜙 is in position of 𝜇 in Proposition 6. Under our assumptions, by Proposition 7, −𝜙 is Lipschitz continuous and the estimate of 𝜕(−𝜙)(𝑥) is 𝜕−𝜙(𝑥)⊂_{𝑦∈𝑌0(𝑥)}𝑢^∗∈𝑋^∗∣𝑢^∗,0∈𝜕−𝑔𝑥,𝑦+^𝑙_𝑖=1𝛼_𝑖𝜕𝑣_𝑖𝑥,𝑦forsome𝛼₁,…,𝛼_𝑙∈ℝ^𝑙₊with𝛼_𝑖𝑣_𝑖𝑥,𝑦=0,𝑖=1,…,𝑙. (37) If 𝑥∈𝑀 solves GSIP, then it also solves min_𝑥𝐹(𝑥). By (31), (32), and (33), there exists 𝜇∈[0,1] such that 0∈𝜇𝜕𝑓(𝑥)−1−𝜇co𝜕−𝜙(𝑥). (38) Combining (37) and (38), there is 𝑦^𝑗∈𝑌₀(𝑥) and 𝜆_𝑗≥0, 𝛼^𝑗_𝑖≥0,  𝑖=1,…,𝑙,  𝑗=1,…,𝑘 such that ∑^𝑘_𝑗=1𝜆_𝑗=1 and 0∈𝜇𝜕𝑓(𝑥)+,01−𝜇^𝑘_𝑗=1𝜆_𝑗−𝜕−𝑔𝑥,𝑦^𝑗−^𝑙_𝑖=1𝛼^𝑗_𝑖𝜕𝑣_𝑖𝑥,𝑦^𝑗. (39) Letting 𝜆₀=𝜇, 𝜆_𝑗=(1−𝜇)𝜆_𝑗, 𝛼^𝑗_𝑖=𝛼^𝑗_𝑖, 𝑖=1,…,𝑙, 𝑗=1,…,𝑘, we obtain the desired result.

Corollary 9. In addition to the assumptions in Theorem 8, assume that 𝑓, 𝑔, and 𝑣 are continuously differentiable. Then the optimality condition at the optimal point 𝑥 is that there exist 𝑦^𝑗∈𝑌₀(𝑥) and 𝜆₀≥0,  𝜆_𝑗≥0,  𝛼^𝑗_𝑖≥0, 𝑖=1,…,𝑙, 𝑗=1,…,𝑘 such that ∑^𝑘_𝑗=1𝜆_𝑗=1 and 0=𝜆₀∇_𝑥𝑓(𝑥)+^𝑘_𝑗=1𝜆_𝑗∇_𝑥𝑔𝑥,𝑦^𝑗−^𝑙_𝑖=1𝛼^𝑗_𝑖∇_𝑥𝑣_𝑖𝑥,𝑦^𝑗,0=^𝑘_𝑗=1𝜆_𝑗∇_𝑦𝑔𝑥,𝑦^𝑗−^𝑙_𝑖=1𝛼^𝑗_𝑖∇_𝑦𝑣_𝑖𝑥,𝑦^𝑗. (40)

Next we consider the case where the lower level value function 𝜙 may fail to be Lipschitz and give estimates for the subdifferentials of 𝜙 and thus further derive the optimality conditions for GSIP. However, it requires to use the Clarke subdifferentials.

Proposition 10. Consider the parametric optimization problem same as (24): 𝜇(𝑥)∶=inf𝜑𝑥,𝑦∣𝑦∈𝐺(𝑥), (41) with corresponding solution mapping 𝑀∶𝑋⇉𝑌. Let 𝑥∈dom𝑀. Assume that the following conditions hold: (i)𝜑 is lower semicontinuous at 𝑥;(ii)𝑀 is 𝜇-inner semicompact at 𝑥 and 𝑀(𝑥) is nonempty and compact;(iii)if (𝑢_𝑖,𝑤_𝑖)∈𝜕^∞𝜑(𝑥,𝑦_𝑖), (𝑣_𝑖,−𝑤_𝑖)∈𝑁_gph𝐺(𝑥,𝑦_𝑖), 𝑦_𝑖∈𝑀(𝑥), 𝑖≤𝑛+1, and ∑_𝑖𝑢_𝑖+𝑣_𝑖=0, then 𝑢_𝑖=𝑣_𝑖=0, 𝑤_𝑖=0;(iv)the cones 𝑁_gph𝐺(𝑥,𝑦) and 𝑁_epi𝜑(𝑥,𝑦,𝜑(𝑥,𝑦)) for all 𝑦∈𝑀(𝑥) are pointed. Then one has the inclusion 𝜕𝜇(𝑥)⊂co_{(𝑦∈𝑀𝑥)}𝑢∣(𝑢,0)∈𝜕𝜑𝑥,𝑦+𝑁_gph𝐺𝑥,𝑦. (42)

Proof (sketch: the definition of 𝜕=clco{𝜕+𝜕^∞}). The proof is divided into two parts. First the set on the right hand of the required inclusion, denoted by Λ, is closed. The second step is to justify that 𝜕+𝜕^∞⊂Λ.
Let ∑^𝑛+1_𝑖=1𝜆^𝑘_𝑖𝑢^𝑘_𝑖→𝑢, 𝜆^𝑘_𝑖≥0, ∑^𝑛+1_𝑖=1𝜆^𝑘_𝑖=1, and 𝑢^𝑘_𝑖,0=𝑥^𝑘_1𝑖,𝑦^𝑘_1𝑖+𝑥^𝑘_2𝑖,𝑦^𝑘_2𝑖,𝑖=1,2,…,𝑛+1, (43) where (𝑥^𝑘_1𝑖,𝑦^𝑘_1𝑖)∈𝜕𝜑(𝑥,𝑦^𝑘_𝑖), (𝑥^𝑘_2𝑖,𝑦^𝑘_2𝑖)∈𝑁_gph𝐺(𝑥,𝑦^𝑘_𝑖), 𝑦^𝑘_𝑖∈𝑀(𝑥), 𝑖=1,…,𝑛+1. We have to show that 𝑢∈Λ. We may assume that 𝑦^𝑘_𝑖→𝑦_𝑖, 𝑖=1,…,𝑛+1, as 𝑀(𝑥) is compact. We show first that the sequence {𝑧^𝑘∶=(𝑧^𝑘_𝑖)_𝑖=(𝑥^𝑘_1𝑖,𝑦^𝑘_1𝑖,𝑥^𝑘_2𝑖,𝑦^𝑘_2𝑖)}_𝑘 is bounded. Suppose on the contrary that ‖𝑧^𝑘‖→∞. Then for each 𝑖, 1𝑧^𝑘𝑢^𝑘_𝑖=1,0𝑧^𝑘𝑥^𝑘_1𝑖,𝑦^𝑘_1𝑖+𝑢^𝑘_2𝑖,𝑣^𝑘_2𝑖,with𝑢^𝑘_2𝑖,𝑣^𝑘_2𝑖=1𝑧^𝑘𝑥^𝑘_2𝑖,𝑦^𝑘_2𝑖. (44) Multiplying (44) by 𝜆^𝑘_𝑖 and taking summation over 𝑖, we have 1𝑧^𝑘𝑛+1_𝑖=1𝜆^𝑘_𝑖𝑥^𝑘_1𝑖,^𝑛+1_𝑖=1𝜆^𝑘_𝑖𝑦^𝑘_1𝑖+^𝑛+1_𝑖=1𝜆^𝑘_𝑖𝑢^𝑘_2𝑖,^𝑛+1_𝑖=1𝜆^𝑘_𝑖𝑣^𝑘_2𝑖⟶(0,0). (45) Note that for each 𝑖, by definition of 𝜕, 1𝑧^𝑘𝑥^𝑘_1𝑖,𝑦^𝑘_1𝑖,−1𝑧^𝑘∈𝑁_epi𝜑𝑥,𝑦^𝑘_𝑖,𝜑𝑥,𝑦^𝑘_𝑖. (46) Let (𝑢^𝑘_1𝑖,𝑣^𝑘_1𝑖)∶=1/‖𝑧^𝑘‖(𝑥^𝑘_1𝑖,𝑦^𝑘_1𝑖)→(𝑢_1𝑖,𝑣_1𝑖). Then by assumption (iv) and Proposition 3 one has (𝑢_1𝑖,𝑣_1𝑖,0)∈𝑁_epi𝜑(𝑥,𝑦_𝑖,𝜑(𝑥,𝑦_𝑖)), and thus (𝑢_1𝑖,𝑣_1𝑖)∈𝜕^∞𝜑(𝑥,𝑦_𝑖). Then 1𝑧^𝑘𝑛+1_𝑖=1𝜆^𝑘_𝑖𝑥^𝑘_1𝑖,^𝑛+1_𝑖=1𝜆^𝑘_𝑖𝑦^𝑘_1𝑖⟶^𝑛+1_𝑖=1𝜆_𝑖𝑢_1𝑖,𝑣_1𝑖, (47) where 𝜆_𝑖∶=lim_𝑘→∞𝜆^𝑘_𝑖, ∑^𝑛+1_𝑖=1𝜆_𝑖=1. Let (𝑢^𝑘_2𝑖,𝑣^𝑘_2𝑖)→(𝑢_2𝑖,𝑣_2𝑖)∈𝑁_gph𝐺(𝑥,𝑦_𝑖). Then 𝑣_1𝑖+𝑣_2𝑖=0 from (43). Combining (45) and (47), we have ^𝑛+1_𝑖=1𝜆_𝑖𝑢_1𝑖,𝑣_1𝑖+𝑢_2𝑖,𝑣_2𝑖=0. (48)
Based on the assumption (iii), we have (𝑢_1𝑖,𝑣_1𝑖)=(0,0),(𝑢_2𝑖,𝑣_2𝑖)=(0,0). This contradicts the fact that (𝑢_1𝑖,𝑣_1𝑖,𝑢_2𝑖,𝑣_2𝑖) is of norm 1, and thus {𝑧^𝑘} is bounded. Then (𝑥^𝑘_1𝑖,𝑦^𝑘_1𝑖),(𝑥^𝑘_2𝑖,𝑦^𝑘_2𝑖) have convergent subsequences, say (𝑥^𝑘_1𝑖,𝑦^𝑘_1𝑖)→(𝑥_1𝑖,𝑦_1𝑖)∈𝜕𝜑(𝑥,𝑦_𝑖),(𝑥^𝑘_2𝑖,𝑦^𝑘_2𝑖)→(𝑥_2𝑖,𝑦_2𝑖)∈𝑁_gph𝐺(𝑥,𝑦_𝑖). Thus 𝑢=∑𝜆_𝑖𝑢_𝑖 with 𝑢_𝑖=𝑥_1𝑖+𝑥_2𝑖. That is to say, Λ is closed.
Next, we justify that 𝜕+𝜕^∞⊂Λ. Assume that 𝑢₁∈𝜕𝜇(𝑥), 𝑢₂∈𝜕^∞𝜇(𝑥). Under the semicompactness assumption, invoking [17, Theorem 1.108], one gets that 𝜕𝜇(𝑥)⊂𝑢∣(𝑢,0)∈∪_{𝑦∈𝑀(𝑥)}𝜕𝜑𝑥,𝑦+𝛿𝑥,𝑦,𝜕,gph𝐺_∞𝜇(𝑥)⊂𝑢∣(𝑢,0)∈∪_{𝑦∈𝑀(𝑥)}𝜕^∞𝜑𝑥,𝑦+𝛿𝑥,𝑦,gph𝐺. (49)
Employing the sum rule from [17, Theorem 3.36] to the two above leads to 𝑢₁+𝑢₂,0∈𝜕𝜑𝑥,𝑦+𝑁_gph𝐺𝑥,𝑦+𝜕^∞𝜑𝑥,𝑦+𝑁_gph𝐺𝑥,𝑦⊂clco𝜕𝜑𝑥,𝑦+𝜕^∞𝜑𝑥,𝑦+𝑁_gph𝐺𝑥,𝑦=𝜕𝜑𝑥,𝑦+𝑁_gph𝐺𝑥,𝑦, (50) which completes the proof.

Theorem 11 (convexified normal cone to inequality system [22]). Consider 𝐺 defined by inequality system 𝐺(𝑥)={𝑦∣𝜙(𝑥,𝑦)≤0}. Let 𝜙 be Lipschitz, and the qualification (non-smooth MFCQ) at (𝑥,𝑦) holds: _{𝑖∈𝐼𝑥,𝑦}𝜆_𝑖𝑤_𝑖=0𝑤𝑖𝑡ℎ𝑤_𝑖∈𝜕𝜙_𝑖𝑥,𝑦,𝜆_𝑖≥0⟹𝜆_𝑖=0𝑓𝑜𝑟𝑖∈𝐼𝑥,𝑦, (51) where 𝐼(𝑥,𝑦)∶={𝑖∣𝜙_𝑖(𝑥,𝑦)=0}. Then 𝑁_gph𝐺𝑥,𝑦⊂_{𝑖∈𝐼𝑥,𝑦}𝜆_𝑖𝜕𝜙_𝑖𝑥,𝑦∣𝜆_𝑖≥0. (52)

As mentioned, GSIP can be relaxed into the following bilevel programming problem: min_𝑥,𝑦𝑓(𝑥)s.t.𝑔𝑥,𝑦≤0,𝑣𝑥,𝑦≤0,𝜙(𝑥)−𝑔𝑥,𝑦≤0. (53) The feasible set of above problem is a subset of the feasible set 𝑀 of (1). Thus, if 𝑥 solves GSIP and 𝜙(𝑥)=0 and 𝑌(𝑥)≠∅, then 𝑥 also solves problem (6). The perturbed version of the above bilevel problem is min_𝑥,𝑦𝑓(𝑥)s.t.𝑔𝑥,𝑦≤0,𝑣𝑥,𝑦≤0,𝜙(𝑥)−𝑔𝑥,𝑦≤𝑢. (54) Problem (6) is said to be partially calm at (𝑥,𝑦) [23] if thereis𝜅>0andaneighborhood𝑉of𝑥,𝑦,0∈ℝ^𝑛×ℝ^𝑚()𝑓(𝑥)(×ℝsuchthatforall𝑥,𝑦,𝑢∈𝑉feasiblefor54wehave−𝑓𝑥)≥−𝜅|𝑢|. (55) Under the partial calmness condition, problem (6) can be transformed into the problem below, for some constant 𝜅>0, min_𝑥,𝑦𝑓(𝑥)+𝜅𝜙(𝑥)−𝑔𝑥,𝑦s.t.𝑔𝑥,𝑦≤0,𝑣𝑥,𝑦≤0. (56)

Theorem 12 (necessary conditions of optimality of GSIP). Let 𝑥 be an optimal solution of GSIP with 𝜙(𝑥)=0, and 𝑌(𝑥)≠∅. Let the data functions 𝑓, 𝑔, and 𝑣 be Lipschitz and the partial calmness condition (55) hold at (𝑥,𝑦) for some 𝑦∈𝑌₀(𝑥). Assume that the following conditions hold: (i) Qualification (51) holds for Ω∶={(𝑥,𝑦)∣𝑣(𝑥,𝑦)≤0,𝑔(𝑥,𝑦)≤0} at (𝑥,𝑦). (ii)𝑌₀ is 𝜙-inner semicompact at 𝑥 and 𝑌₀(𝑥)≠∅. (iii) If (𝑢_𝑖,𝑤_𝑖)∈𝜕^∞𝑔(𝑥,𝑦_𝑖), (𝑣_𝑖,−𝑤_𝑖)∈𝑁_gph𝑌(𝑥,𝑦_𝑖), 𝑦_𝑖∈𝑌₀(𝑥), 𝑖≤𝑛+1, and ∑_𝑖𝑢_𝑖+𝑣_𝑖=0, then 𝑢_𝑖=𝑣_𝑖=0 and 𝑤_𝑖=0. (iv) The cones 𝑁_gph𝑌(𝑥,𝑦) and 𝑁_epi𝑔(𝑥,𝑦,𝑔(𝑥,𝑦)) for all 𝑦∈𝑌₀(𝑥) are pointed. Then there is 𝜅>0, 𝜆_𝑖≥0, 𝑦_𝑖∈𝑌₀(𝑥), 𝑖=1,…,𝑟 such that ∑^𝑟_𝑖=1𝜆_𝑖=1, and 0∈𝜕𝑓(𝑥),0+𝜅^𝑟_𝑖=1𝜆_𝑖𝜕𝑔𝑥,𝑦_𝑖−𝜕𝑔𝑥,𝑦+^𝑟_𝑖=1𝑁_gph𝑌𝑥,𝑦_𝑖+𝑁_Ω𝑥,𝑦. (57)

Proof. Let Ω={(𝑥,𝑦)∣𝑣(𝑥,𝑦)≤0,𝑔(𝑥,𝑦)≤0}. Under our assumptions, GSIP can be relaxed into problem (6) and (𝑥,𝑦) also solves (6) for all 𝑦∈𝑌₀(𝑥). Due to the partial calmness of (6), we also have that (𝑥,𝑦) solves (56). Under the qualification assumption, the necessary optimality condition for problem (56) is (see, e.g., [24, Theorem 5.1] or [22, Theorem 6.2]) 0∈𝜕𝑓(𝑥),0+𝜅𝜕𝜙(𝑥),0+𝜅𝜕−𝑔𝑥,𝑦+𝑁_Ω𝑥,𝑦. (58)
If 𝑣∈𝜕⁺(−𝜙)(𝑥), then −𝑣∈𝜕𝜙(𝑥). Indeed, by definition, 𝜕⁺−𝜙(𝑥)=𝑣∣(−𝑣,1)∈𝑁_{hypo(−𝜙)}(𝑥)=()𝑥,−𝜙𝑣∣−𝑣,−1∈𝑁_epi𝜙(𝑥)⊂()∈𝑥,𝜙𝑣∣−𝑣,−1𝑁_epi𝜙𝑥,𝜙(𝑥). (59) Thus, 𝜕𝜙(𝑥)=−𝜕⁺(−𝜙)(𝑥)⊂𝜕𝜙(𝑥)=−𝜕(−𝜙)(𝑥), and from (58), 0∈𝜕𝑓(𝑥),0−𝜅𝜕−𝜙(𝑥),0+𝜅𝜕−𝑔𝑥,𝑦+𝑁_Ω𝑥,𝑦. (60)
Applying Proposition 10 to −𝜙, there are 𝜆_𝑖≥0, 𝑦_𝑖∈𝑌₀(𝑥),  𝑖=1,…,𝑟 such that ∑^𝑟_𝑖=1𝜆_𝑖=1 and 𝜕−𝜙(𝑥),0⊂^𝑟_𝑖=1𝜆_𝑖𝜕−𝑔𝑥,𝑦_𝑖+𝑁_gph𝑌𝑥,𝑦_𝑖. (61)
So, noting that 𝜕(−𝑔)=−𝜕𝑔, 0∈𝜕𝑓(𝑥),0+𝜅^𝑟_𝑖=1𝜆_𝑖𝜕𝑔𝑥,𝑦_𝑖−𝜅𝜕𝑔𝑥,𝑦+𝜅^𝑟_𝑖=1𝜆_𝑖𝑁_gph𝑌𝑥,𝑦_𝑖+𝑁_Ω𝑥,𝑦⊂𝜕𝑓(𝑥),0+𝜅^𝑟_𝑖=1𝜆_𝑖𝜕𝑔𝑥,𝑦_𝑖−𝜕𝑔𝑥,𝑦+^𝑟_𝑖=1𝑁_gph𝑌𝑥,𝑦_𝑖+𝑁_Ω𝑥,𝑦. (62)
This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgments

This work is supported by the National Science Foundation of China (11001289) and the Key Project of Chinese Ministry of Education (211151).