# Optimality Conditions for Nonsmooth Generalized Semi-Infinite Programs

Authors: Zhangyou Chen; Zhe Chen
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction

Generalized semi-infinite programming problem (GSIP) is of the form(1)minf(x)s.t.nx∈ℝn,g(x,y)≤0,y∈Y(x),
where Y(x):={y∈ℝm∣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,…,λp, not all zero, such that
(2)λ0∇f(x-)+∑j=1p‍λj∇xL(x-,y-j,αj,βj)=0,
where L(x,y,α,β)=αg(x,y)-〈β,v(x,y)〉 and each (αj,βj) is the usual FJ-multiplier of the lower level problem at its optimal solution y-j(3)Q(x-)maxg(x-,y)s.t.xv(x-,y)≤0.
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(4)minf(x)s.t.ϕ(x)≤0,
where ϕ(x) is the optimal value of Q(x). Then we can relate GSIP to the min-max problem
(5)minxmax{f(x)-f(x-),ϕ(x)};
see [3] for more details. On the other hand, GSIP can be related to the following bilevel problem:
(6)min(x,y)f(x)s.t.ng(x,y)≤0,s.t.ny∈arg max{g(x,y):y∈Y(x)}.
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:
(7)min(x,y)f(x)s.t.(x,y)∈Ω,G(x,y)≤0,
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 ℝn, the regular normal cone N^A of A at x-∈A is defined by
(8)N^A(x-)={v∈ℝn∣limsupx→Ax-〈v,x-x-〉∥x-x-∥≤0}.
The (general) normal cone NA of A at x- is defined by
(9)NA(x-)=limsupx→x-N^A(x)={v∣∃xk⟶x-,∃vk⟶vwithvk∈N^A(xk)}.
Given a function f:ℝn→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
(10)∂^f(x-):={u∈ℝn∣liminfx→x-,x≠x-(f(x)-f(x-)-〈u,x-x-〉)∥x-x-∥≥0}.
The general (basic, limiting) and singular subdifferential of f at x- are defined, respectively, by
(11)∂f(x-):=limsupx→fx-∂^f(x),∂∞f(x-):=limsupx→fx-,λ↓0λ∂^f(x).
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-):=limsupx→fx-∂^+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(12)N¯A(x):=T^A(x)°,
where T^A(x)=liminfy→Ax,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
(13)N¯A(x):=cl coNA(x).
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 off at x are defined by
(14)∂¯f(x):={v∣(v,-1)∈N¯epif(x,f(x))},∂¯∞f(x):={v∣(v,0)∈N¯epif(x,f(x))}.
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]).
Letf be proper and lsc around x-∈domf. Then,
(15)∂¯f(x-)=clco[∂f(x-)+∂∞f(x-)].
If, in particular, f is Lipschitz continuous at x-, then
(16)∂¯f(x-)=clco∂f(x-).The normal coneNA enjoys the robustness property NA(x-)=limsupx→x-NA(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
(17)A:={x∈ℝ3∣x3=x1x2orx3=-x1x2},(18)x-=(0,0,0).
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]).
IfA is convex, or if N¯A(x-) is pointed, then the multifunction N¯A is closed at x-; that is, for all xk→x-, yk→N¯A(xk)y-, one has y-∈N¯A(x-).Proposition 3.
The Clarke normal cone has the robustness property(19)N¯A(x)=limsupy→xN¯A(y)
provided that NA(x) is pointed.Proof.
It suffices to prove thatlimsupy→xcoNA(y)⊂cl colimsupy→xNA(y)=N¯A(x). Let v∈limsupy→xcoNA(y). Then there are yk∈A, vik∈NA(yk), i=1,…,n+1 such that
(20)∑i=1n+1‍vik⟶vask⟶∞,
since the sets NA(yk) are cones. Let λk=∑i=1n+1‍∥vik∥. Then {λk} is bounded; that is also to say, {vik} are bounded for all i. Otherwise, ∑i=1n+1‍vik/λk→0. That is, v1+⋯+vn+1=0, where vi is the limit of {vik/λk}k for each i. Note that vi∈NA(x) since NA(x)=limsupy→xNA(y). Thus v1=⋯=vn+1=0 by the pointedness of NA(x). On the other hand, ∑i‍∥vi∥=1. This is a contradiction. Thus the sequence {vik} is bounded. By taking subsequences, we may assume that vik→vi. Then vi∈NA(x) and v=v1+⋯+vn+1. This completes the proof.The following definitions are required for further development.Definition 4 (see [17, Definition  1.63]).
LetS:X⇉Y be a set-valued mapping with x-∈domS, the domain of S. (i)
Giveny-∈S(x-), we say that the mapping S is inner semicontinuous at (x-,y-) if for every sequence xk→x- there is a sequence yk∈S(xk) converging to y- as k→∞.(ii)
S is inner semicompact at x- if for every sequence xk→x- there is a sequence yk∈S(xk) 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, xk→x- is replaced by xk→x- with μ(xk)→μ(x-).Here the concept ofμ-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(21)(maxϕi)(x)=max{ϕi(x)∣i=1,…,l}.
Let ϕi be lower semicontinuous around x- for i∈I(x-):={i∣max(ϕi)(x-)=ϕi(x-)} and be upper semicontinuous at x- for i∉I(x-). Assume that the qualification holds:
(22)[∑i∈I(x-)‍xi*=0,xi*∈∂∞ϕi(x-)]⟹xi*=0,i∈I(x-).
Then
(23)∂(maxϕi)(x-)⊂⋃‍{∑i∈I(x-)‍λi∂ϕi(x-)∣(λ1,…,λl)∈Λ(x-)},
where Λ(x-)={(λ1,…,λl)∣λi≥0,∑i=1l‍λi=1,λi(ϕi(x-)-(maxϕi)(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(24)μ(x):=inf{φ(x,y)∣y∈G(x)},
and let M(x):={y∈G(x)∣μ(x)=φ(x,y)}, G(x):={y∈ℝm∣φi(x,y)≤0,i=1,…,l}. For simplicity, one does not  consider the case with equality constraints involved.  (i)
Assume thatMisμ-inner semicontinuous at(x-,y-)∈gphM(the graph ofM), thatφand allφiare Lipschitz continuous around(x-,y-), and that the following qualification condition is satisfied:(25)onlythevector(λ1,…,λl)=0∈ℝlsatisfiestherelation0∈∑i=1l‍λi∂φi(x-,y-)forsome(λ1,…,λl)∈ℝ+lwithλiφi(x-,y-)=0,i=1,…,l.
One has the inclusions(26)∂μ(x-)⊂{∑i=1l‍u*∈X*∣(u*,0)∈∂φ(x-,y-)+∑i=1l‍λi∂φi(x-,y-)forsome(λ1,…,λl)∈ℝ+lwithλiφi(x-,y-)=0,i=1,…,l∑i=1l‍}.(ii)
Assume thatMisμ-inner semicompact atx-, and allφandφiare Lipschitz continuous around(x-,y-)for ally-∈M(x-), and qualification (25) holds for all(x-,y-),y-∈M(x-). Then(27)∂μ(x-)⊂⋃y-∈M(x-)‍{∑i=1l‍u*∈X*∣(u*,0)∈∂φ(x-,y-)+∑i=1l‍λi∂φi(x-,y-),forsome(λ1,…,λl)∈ℝ+mwithλiφi(x-,y-)=0,i=1,…,l∑i=1l‍}.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 thatMisμ-inner semicontinuous at(x-,y-)∈gphMandφis locally Lipschitz around this point. Thenμis Lipschitz aroundx-provided that it islscaroundx-andGis Lipschitz-like around(x-,y-).(ii)
Assume thatMisμ-inner compact atx-andφis locally Lipschitz around(x-,y-)for ally-∈M(x-). Thenμis Lipschitz aroundx-provided that it islscaroundx-andGis Lipschitz-like around(x-,y-)for ally-∈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
(28)minxmax{f(x)-f(x-),ϕ(x)},
where ϕ(x):=supy∈Y(x)g(x,y). Let Y0(x):={y∈Y(x):ϕ(x)=g(x,y)}. Denote by
(29)F(x)=max{f(x)-f(x-),ϕ(x)}.
If x- solves GSIP (1), then x- also solves problem
(30)min{F(x):x∈ℝn},
and thus by generalized Fermat's rule (cf. [16, Theorem 10.1]), we have
(31)0∈∂F(x-).
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)
(32)0∈μ∂f(x-)+(1-μ)∂ϕ(x-)⊂μ∂f(x-)+(1-μ)co∂ϕ(x-).
Note that for a Lipschitz function ϕ,
(33)co∂ϕ(x-)=-co∂(-ϕ)(x-).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 λ-0≥0,  λ-j≥0,  α-ij≥0, i=1,…,l, j=1,…,k such that ∑j=1k‍λ-j=1 and
(34)0∈(λ-0∂f(x-),0)+∑j=1k‍λ-j[-∂(-g)(x-,y-j)-∑i=1l‍α-ij∂vi(x-,y-j)].
If in addition g and all components of v are regular at all (x-,y-j), then the optimality is of the form
(35)0∈λ-0∂f(x-)+∑j=1k‍λ-j[-∂x(-g)(x-,y-j)-∑i=1l‍α-ij∂xvi(x-,y-j)],0∈∑j=1k‍λ-j[-∂y(-g)(x-,y-j)-∑i=1l‍α-ij∂yvi(x-,y-j)].
Note that -∂(-g)=∂+g.Proof.
Under regularity and Lipschitz continuity, since the following calculus rule for basic subgradients holds (see [16, Corollary  10.11]):
(36)∂xf(x,y)={u∣∃vs.t.(u,v)∈∂f(x,y)},
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
(37)∂(-ϕ)(x-)⊂⋃y-∈Y0(x-)‍{∑i=1l‍u*∈X*∣(u*,0)∈∂(-g)(x-,y-)+∑i=1l‍αi∂vi(x-,y-)for  some(α1,…,αl)∈ℝ+lwithαivi(x-,y-)=0,i=1,…,l∑i=1l‍}.
If x-∈M solves GSIP, then it also solves minxF(x). By (31), (32), and (33), there exists μ∈[0,1] such that
(38)0∈μ∂f(x-)-(1-μ)co∂(-ϕ)(x-).
Combining (37) and (38), there is y-j∈Y0(x-) and λj≥0, αij≥0,  i=1,…,l,  j=1,…,k such that ∑j=1k‍λj=1 and
(39)0∈(μ∂f(x-),0)+(1-μ)∑j=1k‍λj[-∂(-g)(x-,y-j)-∑i=1l‍αij∂vi(x-,y-j)].
Letting λ-0=μ, λ-j=(1-μ)λj, α-ij=αij, i=1,…,l, j=1,…,k, we obtain the desired result.Corollary 9.
In addition to the assumptions in Theorem8, 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 λ-0≥0,  λ-j≥0,  α-ij≥0, i=1,…,l, j=1,…,k such that ∑j=1k‍λ-j=1 and
(40)0=λ-0∇xf(x-)+∑j=1k‍λ-j[∇xg(x-,y-j)-∑i=1l‍α-ij∇xvi(x-,y-j)],0=∑j=1k‍λ-j[∇yg(x-,y-j)-∑i=1l‍α-ij∇yvi(x-,y-j)].Next we consider the case where the lower level value functionϕ 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):
(41)μ(x):=inf{φ(x,y)∣y∈G(x)},
with corresponding solution mapping M:X⇉Y. Let x-∈domM. Assume that the following conditions hold: (i)
φis lower semicontinuous atx-;(ii)
Misμ-inner semicompact atx-andM(x-)is nonempty and compact;(iii)
if(ui,wi)∈∂-∞φ(x-,y-i),(vi,-wi)∈N¯gphG(x-,y-i),y-i∈M(x-),i≤n+1, and∑i‍ui+vi=0, thenui=vi=0,wi=0;(iv)
the conesNgphG(x-,y-)andNepiφ(x-,y-,φ(x-,y-))for ally-∈M(x-)are pointed.Then one has the inclusion
(42)∂-μ(x-)⊂co(⋃y-∈M(x-)‍{u∣(u,0)∈∂-φ(x-,y-)+N¯gphG(x-,y-)}).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 byΛ, is closed. The second step is to justify that ∂+∂∞⊂Λ.
Let∑i=1n+1‍λikuik→u, λik≥0, ∑i=1n+1‍λik=1, and
(43)(uik,0)=(x1ik,y1ik)+(x2ik,y2ik),i=1,2,…,n+1,
where (x1ik,y1ik)∈∂-φ(x-,y-ik), (x2ik,y2ik)∈N¯gphG(x-,y-ik), y-ik∈M(x-), i=1,…,n+1. We have to show that u∈Λ. We may assume that y-ik→y-i, i=1,…,n+1, as M(x-) is compact. We show first that the sequence {zk:=(zik)i=(x1ik,y1ik,x2ik,y2ik)}k is bounded. Suppose on the contrary that ∥zk∥→∞. Then for each i,
(44)1∥zk∥(uik,0)=1∥zk∥(x1ik,y1ik)+(u2ik,v2ik),with(u2ik,v2ik)=1∥zk∥(x2ik,y2ik).
Multiplying (44) by λik and taking summation over i, we have
(45)1∥zk∥(∑i=1n+1‍λikx1ik,∑i=1n+1‍λiky1ik)+(∑i=1n+1‍λiku2ik,∑i=1n+1‍λikv2ik)⟶(0,0).
Note that for each i, by definition of ∂-,
(46)(1∥zk∥(x1ik,y1ik),-1∥zk∥)∈N¯epiφ(x-,y-ik,φ(x-,y-ik)).
Let (u1ik,v1ik):=1/∥zk∥(x1ik,y1ik)→(u1i,v1i). Then by assumption (iv) and Proposition 3 one has (u1i,v1i,0)∈N¯epiφ(x-,y-i,φ(x-,y-i)), and thus (u1i,v1i)∈∂-∞φ(x-,y-i). Then
(47)1∥zk∥(∑i=1n+1‍λikx1ik,∑i=1n+1‍λiky1ik)⟶∑i=1n+1‍λi(u1i,v1i),
where λi:=limk→∞λik, ∑i=1n+1‍λi=1. Let (u2ik,v2ik)→(u2i,v2i)∈N¯gphG(x-,y-i). Then v1i+v2i=0 from (43). Combining (45) and (47), we have
(48)∑i=1n+1‍λi((u1i,v1i)+(u2i,v2i))=0.
Based on the assumption (iii), we have(u1i,v1i)=(0,0),(u2i,v2i)=(0,0). This contradicts the fact that (u1i,v1i,u2i,v2i) is of norm 1, and thus {zk} is bounded. Then (x1ik,y1ik),(x2ik,y2ik) have convergent subsequences, say (x1ik,y1ik)→(x1i,y1i)∈∂-φ(x-,y-i),(x2ik,y2ik)→(x2i,y2i)∈N¯gphG(x-,y-i). Thus u=∑‍λiui with ui=x1i+x2i. 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
(49)∂μ(x-)⊂{u∣(u,0)∈∪y-∈M(x-)∂(φ(x-,y-)+δ((x-,y-),gphG))},∂∞μ(x-)⊂{u∣(u,0)∈∪y-∈M(x-)∂∞(φ(x-,y-)+δ((x-,y-),gphG))}.
Employing the sum rule from [17, Theorem 3.36] to the two above leads to
(50)(u1+u2,0)∈[∂φ(x-,y-)+NgphG(x-,y-)]+[∂∞φ(x-,y-)+NgphG(x-,y-)]⊂cl co[∂φ(x-,y-)+∂∞φ(x-,y-)]+N¯gphG(x-,y-)=∂-φ(x-,y-)+N¯gphG(x-,y-),
which completes the proof.Theorem 11 (convexified normal cone to inequality system [22]).
ConsiderG 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:
(51)[∑i∈I(x-,y-)‍λiwi=0withwi∈∂ϕi(x-,y-),λi≥0]⟹λi=0fori∈I(x-,y-),
where I(x-,y-):={i∣ϕi(x-,y-)=0}. Then
(52)N¯gphG(x-,y-)⊂{∑i∈I(x-,y-)‍λi∂-ϕi(x-,y-)∣λi≥0}.As mentioned, GSIP can be relaxed into the following bilevel programming problem:(53)min(x,y)f(x)s.t.ng(x,y)≤0,v(x,y)≤0,ϕ(x)-g(x,y)≤0.
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
(54)min(x,y)f(x)s.t.ng(x,y)≤0,v(x,y)≤0,ϕ(x)-g(x,y)≤u.
Problem (6) is said to bepartially calmat (x-,y-) [23] if
(55)thereisκ>0andaneighborhoodVof(x-,y-,0)∈ℝn×ℝm×ℝsuchthatforall(x,y,u)∈Vfeasiblefor(54)wehavef(x)-f(x-)≥-κ|u|.
Under the partial calmness condition, problem (6) can be transformed into the problem below, for some constant κ>0,
(56)min(x,y)f(x)+κ[ϕ(x)-g(x,y)]s.t.ng(x,y)≤0,v(x,y)≤0.Theorem 12 (necessary conditions of optimality of GSIP).
Letx- 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(ui,wi)∈∂-∞g(x-,y-i), (vi,-wi)∈N¯gphY(x-,y-i), y-i∈Y0(x-), i≤n+1, and ∑i‍ui+vi=0, then ui=vi=0 and wi=0.(iv)
The conesNgphY(x-,y) and Nepig(x-,y,g(x-,y)) for all y∈Y0(x-) are pointed. Then there is κ>0, λi≥0, y-i∈Y0(x-), i=1,…,r such that ∑i=1r‍λi=1, and
(57)0∈(∂-f(x-),0)+κ∑i=1r‍λi[∂-g(x-,y-i)-∂-g(x-,y-)]+∑i=1r‍N¯gphY(x-,y-i)+N¯Ω(x-,y-).Proof.
LetΩ={(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])
(58)0∈(∂f(x-),0)+κ(∂ϕ(x-),0)+κ∂(-g)(x-,y-)+NΩ(x-,y-).
Ifv∈∂+(-ϕ)(x), then -v∈∂-ϕ(x). Indeed, by definition,
(59)∂+(-ϕ)(x)={v∣(-v,1)∈Nhypo(-ϕ)(x,-ϕ(x))}={v∣(-v,-1)∈Nepiϕ(x,ϕ(x))}⊂{v∣(-v,-1)∈N¯epiϕ(x,ϕ(x))}.
Thus, ∂ϕ(x)=-∂+(-ϕ)(x)⊂∂-ϕ(x)=-∂-(-ϕ)(x), and from (58),
(60)0∈(∂-f(x-),0)-κ(∂-(-ϕ)(x-),0)+κ∂-(-g)(x-,y-)+N¯Ω(x-,y-).
Applying Proposition10 to -ϕ, there are λi≥0, y-i∈Y0(x-),  i=1,…,r such that ∑i=1r‍λi=1 and
(61)(∂-(-ϕ)(x-),0)⊂∑i=1r‍λi[∂-(-g)(x-,y-i)+N¯gphY(x-,y-i)].
So, noting that∂-(-g)=-∂-g,
(62)0∈(∂-f(x-),0)+κ∑i=1r‍λi∂-g(x-,y-i)-κ∂-g(x-,y-)+κ∑i=1r‍λiN¯gphY(x-,y-i)+N¯Ω(x-,y-)⊂(∂-f(x-),0)+κ∑i=1r‍λi[∂-g(x-,y-i)-∂-g(x-,y-)]+∑i=1r‍N¯gphY(x-,y-i)+N¯Ω(x-,y-).
This completes the proof.



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*Source: 131938-2013-09-29.xml*