JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 118, No. 1, pp. 117–133, July 2003 ( 2003)

A Practical Optimality Condition withoutConstraint Qualifications forNonlinear Programming1,2

J. M. MARTINEZ3

AND B. F. SVAITER4

Communicated by G. Di Pillo

Abstract. A new optimality condition for minimization with generalconstraints is introduced. Unlike the KKT conditions, the new con-dition is satisfied by local minimizers of nonlinear programming prob-lems, independently of constraint qualifications. The new condition isstrictly stronger than and implies the Fritz–John optimality conditions.Sufficiency for convex programming is proved.

Key Words. Optimality conditions, Karush–Kuhn–Tucker conditions,minimization algorithms, constrained optimization.

1. Introduction

In the global convergence theory of most nonlinear programming algo-rithms, it is proved that some limit point (perhaps all the limit points) ofthe generated sequence satisfies a necessary condition for constrained mini-mization. Usually, the necessary condition says that the Karush–Kuhn–Tucker conditions are satisfied if the limit point satisfies some constraintqualification. When the constraint qualification is the one introduced byMangasarian and Fromovitz (Ref. 1), we say that the Fritz–John optimalitycondition is satisfied. Perhaps separation techniques are the most powerfultools for proving optimality conditions; see Refs. 2–6. Therefore, according

1This work was supported by FAPESP Grant PT 2001-04597-4, by CNPq Grant 30120093-9(RN), and by FAPERJ, PRONEX-Optimization, and FAEP-UNICAMP.

2We acknowledge Alfredo Iusem for his helpful comments on the first draft of this paper. Weare also indebted to two anonymous referees and the Associate Editor for helpful suggestions.

3Professor, Department of Applied Mathematics, Institute of Mathematics, University of Cam-pinas, Campinas, SP, Brazil.

4Professor, Institute of Pure and Applied Mathematics (IMPA), CNPq, Rio de Janeiro, Brazil.

1170022-3239030700-01170 2003 Plenum Publishing Corporation

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to that model global theorem, the algorithmic sequence might converge topoints where the constraint qualification is not satisfied. This means that apotential failure of the nonlinear programming algorithm could be due tothe presence of nonregular feasible attraction points that are not localminimizers.

However, convergence of good algorithms to nonregular feasible pointsthat are not local minimizers is not frequent. This is not surprising, sincealgorithms are designed taking into account, at every iteration, the necessityof minimizing an objective function. So, convergence to points at which theobjective function has no special properties should be unusual, at least forwell-designed methods. On the other hand, convergence to local minimizerscan be guaranteed only under restrictive second-order properties of the con-straints (see Ref. 7) or convexity-like assumptions.

Therefore, both practical experience and common sense indicates thatthe set of feasible limit points of good algorithms for constrained optimiz-ation, although being larger than the set of local minimizers, does notinclude the set of all nonregular feasible points. For example, 0 is a non-regular point of the problem of minimizing x subject to A1⁄x3⁄0, but noreasonable minimization algorithm will converge to it.

The observations above lead us to seek optimality conditions that fitbetter the behavior of practical methods. Another motivation for dis-covering sharper than Fritz–John optimality conditions comes from con-sidering mathematical programming with equilibrium constraints (MPEC)problems; see Refs. 8–10 and references therein. In standard formulationsof MPEC, no feasible point satisfies the Mangasarian–Fromovitz constraintqualification; thus, all the feasible points satisfy the Fritz–John optimalitycondition. So, it is necessary to study optimality conditions that approxi-mate the minimizers of the problem more accurately than the Fritz–Johnconditions.

In this paper, we introduce an optimality condition that, roughly for-mulated, says that an approximate gradient projection tends to zero. Forthis reason, we call it the approximate gradient projection (AGP) property.We prove that the AGP property is satisfied by local minimizers of con-strained optimization problems independently of constraint qualificationsand using only first-order differentiability. Therefore, the AGP propertyis a genuine necessary optimality condition that does not use constraintqualifications at all. Moreover, we show that the set of AGP points is con-tained and can be strictly contained in the set of Fritz–John points.

The reputation of the KKT conditions lies not only in being necessaryoptimality conditions under constraint qualifications, but also in the factthat they are sufficient optimality conditions if the nonlinear program isconvex. Essentially, we prove that this is also true for the AGP property.

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This paper is organized as follows. In Section 2, we state rigorously theAGP property; we prove that it is satisfied by every local minimizer of anonlinear programming problem and that it implies the Fritz–John con-dition. In Section 3, we prove that the new condition is sufficient for convexproblems. In Section 4, we define a strong AGP condition and we prove itsmain properties. Conclusions are given in Section 5.

2. Necessary Condition

We consider the nonlinear programming problem

min f (x), s.t.x∈Ω, (1)

where

ΩGx∈ng(x)⁄0, h(x)G0, (2)

f : n→, h: n→m, g: n→p, and all the functions have continuous firstpartial derivatives.

Let γ ∈[−S, 0]. For all x∈n, we define Ω(x, γ ) as the set of pointsz∈n that satisfy

gi (x)Cg′i (x)(zAx)⁄0, if γFgi (x)F0, (3)

g′i (x)(zAx)⁄0, if gi (x)¤0, (4)

h′(x)(zAx)G0. (5)

The set Ω(x, γ ), a closed and convex polyhedron, can be interpreted as alinear approximation of the set of points z∈n that satisfy

h(z)Gh(x),

gi (z)⁄gi (x), if gi (x)¤0,

gi (z)⁄0, if gi (x)∈(γ , 0).

Observe that

Ω(x, 0)Gz∈nh′(x)(zAx)G0, g′i (x)(zAx)⁄0, if gi (x)¤0, iG1, . . . , p.

For all x∈n, we define

d(x, γ )GPΩ(x, γ )(xA∇f (x))Ax, (6)

where PC (y) denotes the orthogonal projection of y onto C for ally∈ n, C⊂n closed, convex. The vector d(x, γ ) will be called the approxi-mate gradient projection.

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We will denote · G · 2 and

B (x, ρ)Gz∈n zAx⁄ρ,

for all x∈n, ρH0. As usual, for all û∈n, ûG(û1 , . . . , ûn), we denote

ûCG(max0, û1, . . . , max 0, ûn).

The main result of this section is proved below. It says that, if x* is alocal minimizer of (1)–(2), we can find points with arbitrarily small approxi-mate gradient projections that are arbitrarily close to x*. The technique ofthis proof is similar to the one used by Bertsekas (Proposition 3.3.5 of Ref.11) to prove the Fritz–John conditions.

Theorem 2.1. Assume that x* is a local minimizer of (1)–(2), and letγ ∈[−S, 0], (, δH0 be given. Then, there exists x∈n such that

xAx*⁄δ and d(x, γ ) ⁄(.

Proof. Let ρ∈(0, δ ) be such that x* is a global minimizer of f (x) on

Ω ∩B (x*, ρ). For all x∈n, define

ϕ(x)Gf (x)C((2ρ) xAx*2.

Clearly, x* is the unique global solution of

min ϕ(x), s.t.x∈Ω ∩B (x*, ρ). (7)

For all x∈ n, µH0, define

Φµ (x)Gϕ(x)C(µ2)[h(x) 2Cg(x)C 2].

The external penalty theory (see for instance Ref. 12) guarantees that, forµ sufficiently large, there exists a solution of

minΦµ (x), s.t. x∈B (x*, ρ), (8)

that is as close as desired to the global minimizer of ϕ(x) onΩ ∩B (x*, ρ). So, for µ large enough, there exists a solution xµ of (8) inthe interior of B (x*, ρ). Therefore,

∇Φµ (xµ)G0.

Thus, writing for simplicity xGxµ , we obtain

0G∇Φµ (x)

G∇ϕ(x)Cµh′(x)Th(x)C ∑gi (x)H0

gi (x)∇gi (x) .

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So,

∇f (x)Cµh′(x)Th(x)C ∑gi (x)H0

gi (x)∇gi (x)C((ρ)(xAx*)G0. (9)

Since xGxµ lies in the interior of the ball, we have that

xAx*FρFδ .

So, by (9),

∇f (x)Cµh′(x)Th(x)C ∑gi (x)H0

gi (x)∇gi (x)⁄(.

So,

[xA∇f (x)]AxCµh′(x)Th(x)C ∑gi (x)H0

gi (x)∇gi (x)⁄(.

Taking projections onto Ω(x, γ ), this implies that

PΩ(x, γ )(xA∇f (x))APΩ(x, γ )xCµh′(x)Th(x)C ∑gi (x)H0

gi (x)∇gi (x)⁄(.

(10)

It remains to prove that

PΩ(x, γ )xCµh′(x)Th(x)C ∑gi (x)H0

gi (x)∇gi (x)Gx.

To see that this is true, consider the convex quadratic subproblem

miny

yAxCµh′(x)Th(x)C ∑gi (x)H0

gi (x)∇gi (x)2

,

s.t. y∈Ω(x, γ ),

and observe that yGx satisfies the sufficient KKT optimality conditionswith the multipliers

λG−µh(x),

νiGµgi (x), for gi (x)¤0,

νiG0, else.

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So, by (10),

PΩ(x, γ )(xA∇f (x))Ax⁄(,

as we wanted to prove.

The following corollary states the AGP property in a closer way to thealgorithmic framework: given a local minimizer, a sequence that satisfiesxk→x* and d(xk, γ )→0 necessarily exists. As we mentioned in the introduc-tion, this property is satisfied naturally by many nonlinear programmingalgorithms (Refs. 12–14), xk being the algorithmic sequence.

Corollary 2.1. If x* is a local minimizer of (1)–(2) and if γ ∈[−S, 0],there exists a sequence xk⊂n such that lim xkGx* and lim d(xk, γ )G0.

Remark 2.1. In general, the identity d(x*, γ )G0 does not hold at alocal minimizer x*. For example, in the problem of minimizing x subject tox2⁄0, we have that

d(x*, γ ) ≠ 0, for all γF0.

In the following, we prove that the AGP conditions are equivalent forall γF0. This property is not true for γ G0. In fact, consider the problemof minimizing x subject to x¤0, with

xkG1k, ∀kG0, 1, 2, . . ..

Clearly,

d(xk, γ )→0, for all γF0,

but d(xk, 0) does not tend to zero.

Property 2.1. Assume that g(x*)⁄0, h(x*)G0, xk→x*, and that, forsome γ ∈[−S, 0), d(xk, γ )→0. Then, d(xk, γ ′ )→0 for all γ ′F0.

Proof. Consider the problems

min xkA∇f (xk)Ay2, (11)

s.t. g′i (xk)(yAxk)⁄0, if gi (xk)¤0, (12)

g′i (xk)(yAxk)⁄0, if 0Hgi (xk)¤γ , (13)

h′(xk)(yAxk)G0 (14)

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and

min xkA∇f (xk)Ay2, (15)

s.t. g′i (xk)(yAxk)⁄0, if gi (xk)¤0, (16)

g′i (xk)(yAxk)⁄0, if 0Hgi (xk)¤γ ′, (17)

h′(xk)(yAxk)G0. (18)

Let yk be the solution of (11)–(14). By the hypothesis, we know thatykAxk→0. Therefore, yk→x*. Let us show that, for k large enough, yk

satisfies the constraints (16)–(18). In the case of (16) and (18), this is obvi-ous. Let us prove that the same is true for (17). We consider two cases:gi (x*)G0 and gi (x*)F0.

If gi (x*)G0, then for k large enough, gi (xk)Hγ . Therefore, if in

addition gi (xk)F0, the constraint (17) is satisfied.

If gi (x*)F0, then since ykAxk→0, we have that, for k large enough,

gi (xk)Cg′i (xk)(ykAxk)F0. (19)

Therefore, the constraints (17) are also satisfied at yk.But yk is a KKT point of the problem (11)–(14). To prove that it is

also a KKT point and thus a solution of (15)–(18), it remains only to provethat the active constraints of (15)–(18) are necessarily constraints that occurin (11)–(14). Again, this is obvious in the case of (16) and (18). By theanalysis performed above, an active constraint of type (17) can correspondonly to gi (x*)G0 and gi (x

k)F0. In this case, gi (xk)Hγ for k large enough;

thus, the constraint is present in the set (13). This completes the proof.

We finish this section proving that the AGP condition implies theFritz–John optimality condition. As in the case of Theorem 2.1, the tech-nique of proof is similar to the one of Proposition 3.3.5 of Ref. 11. Let usrecall the equivalent formulation of the Mangasarian–Fromovitz constraintqualification; see Ref. 15.

Mangasarian–Fromovitz (MF) Constraint Qualification. If λ∈m andµ∈p, µ¤0 are such that

h′(x*)TλCg′(x*)TµG0 and µTg(x*)G0,

then λG0 and µG0.

Theorem 2.2. Assume that x* is a feasible point. Let γ ∈[−S, 0]. Sup-pose that there exists a sequence xk→x* such that d(xk, γ )→0. Then, x* isa Fritz–John point of (1)–(2).

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Proof. Define

ykGPΩ(xk, γ )(xkA∇f (xk)).

So, yk solves

min (12) yAxk2C∇f (xk)T( yAxk),

s.t. y∈Ω(xk, γ ).

Therefore, there exist λ k∈m, µk∈p, µk¤0 such that

∇f (xk)C(ykAxk)Ch′(xk)Tλ kCg′(xk)TµkG0, (20a)

µki [gi (x

k)Cg′i (xk)(ykAxk)]G0, if γFgi (xk)F0, (20b)

µki [g′i (xk)(ykAxk)]G0, if gi (x

k)¤0, (20c)

µki G0, else. (20d)

Moreover, if gi (x*)F0, we have that gi (xk)F0 for k large enough; and since

ykAxk→0, we have also that

gi (xk)Cg′i (xk)(ykAxk)F0

in that case. Therefore, we can assume that

µki G0 whenever gi (x*)F0. (21)

To prove the Fritz–John condition is equivalent to proving that the MFcondition implies the KKT condition. So, from now on, we are going toassume that x* satisfies the Mangasarian–Fromovitz (MF) constraintqualification.

Suppose by contradiction that (λ k, µk) is unbounded. Define, for eachk,

MkG (λ k, µk) SGmaxλ kS , µkS.

Then,

lim sup MkGS.

Refining the sequence (λ k, µk) and reindexing it, we may suppose thatMkH0 for all k and

limk

MkG+S.

Now, define

λ kG(1Mk)λ k, µkG(1Mk)µk.

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Observe that, for all k,

(λ k, µk) SG1.

Hence, the sequence (λ k, µk) is bounded and has a cluster point (λ , µ)satisfying

µ¤0, (λ , µ) SG1.

Dividing (20) by Mk , we obtain

(1Mk)[∇f (xk)C(ykAxk)]Ch′(xk)Tλ kCg′(xk)TµkG0.

Taking the limit along the appropriate subsequence, we conclude that

h′(x*)TλCg′(x*)TµG0.

Together with (21), this contradicts the constraint qualification MF.Now, since in (20), λk and µk are bounded, extracting a convergent

subsequence, we have that

λ k→λ* and µk→µ*¤0.

By (21),

g(x*)Tµ*G0

and, taking limits in (20), the KKT condition follows.

The following corollary states that, under the Mangasarian–Fromovitzconstraint qualification, the AGP condition implies the standard KKT con-ditions of nonlinear programming. It is a trivial consequence of Theorem2.2.

Corollary 2.2. If the hypotheses of Theorem 2.2 hold and x* satisfiesthe MF condition, then the KKT necessary conditions are fulfilled at x*.

Remark 2.2. We showed above that the AGP condition together withthe Mangasarian–Fromovitz constraint qualification implies the KKT con-ditions. This is not true if the constraint qualification does not take place.For example, consider the problem

min x,

s.t. x2G0.

The obvious solution of this problem does not satisfy the KKT conditions.On the other hand, it satisfies the AGP condition. Observe, for example,that the sequence 1k trivially satisfies the AGP property. In fact,

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Theorem 2.1 proves that every local minimizer of a nonlinear programmingproblem satisfies the AGP property.

The set of Fritz–John points can be strictly larger than the set of pointsthat satisfy the AGP condition. Consider the nonlinear program given by

min x,

s.t. x3⁄0.

Take x*G0. Of course, this is not a solution of the problem. Clearly, x*satisfies the Fritz–John condition; but for any sequence xk that convergesto x*, d(xk, η) does not tend to zero. Of course, x* does not satisfy theMangasarian–Fromovitz constraint qualification.

3. Sufficiency in the Convex Case

As it is well known, in convex problems, the KKT conditions implyoptimality. In this section, we prove that essentially the same sufficiencyproperty is true for the AGP optimality condition.

Theorem 3.1. Suppose that, in the nonlinear program (1)–(2), f and gi

are convex, iG1, . . . , p, and h is an affine function. Let γ ∈[−S, 0]. Supposethat x*∈Ω and that xk⊂n are such that lim xkGx*, h(xk)G0 for allkG0, 1, 2, . . . , and lim d(xk, γ )G0. Then, x* is a minimizer of (1)–(2).

Proof. Let us prove first that

Ω⊂Ω(xk, γ ), for all xk∈n.

Assume that z∈Ω. If gi (z)⁄0 and gi (xk)F0, by the convexity of gi we have

that

0¤gi (z)¤gi (xk)Cg′i (xk)(zAxk). (22)

Moreover, if gi (z)⁄0 and gi (xk)¤0,

0¤gi (z)

¤gi (xk)Cg′i (xk)(zAxk)

¤g′i (xk)(zAxk). (23)

Therefore, by (22) and (23), z∈Ω implies that z∈Ω(xk, γ ); so,Ω⊂Ω(xk, γ ). Note that (5) holds since h(xk)Gh(z)G0 and h is affine.

Let us define now

ykGPΩ(xk, γ )(xkA∇f (xk)).

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Let z∈Ω be arbitrary. Since z∈Ω(xk, γ ) and yk minimizesxkA∇f (xk)Ay2 on this set, we have that

∇f (xk)T(zAxk)C(12) zAxk2

¤∇f (xk)T( ykAxk)C(12) ykAxk2.

Taking limits on both sides of the above inequality and using the fact thatykAxk→0, we obtain

∇f (x*)T(zAx*)C(12) zAx*2¤0.

Since Ω is convex, the above inequality holds replacing z withx*Ct(zAx*) for all t∈[0, 1]; so,

t∇f (x*)T(zAx*)Ct2(12) zAx*2¤0.

Thus,

∇f (x*)T(zAx*)C(t2) zAx*2¤0,

for all t∈(0, 1]. Taking limits for t→0, we obtain that

∇f (x*)T(zAx*)¤0.

Since z∈Ω is arbitrary, convexity implies that x* is minimizer of (1)–(2).

Corollary 3.1. Assume that f and gi , iG1, . . . , p, are as in Theorem3.1 and that there are no equality constraints at all. Suppose thatlim xkGx* and d(xk, γ )→0. Then, x* is a minimizer of (1)–(2).

Remark 3.1. Theorem 3.1 is not true if we do not assume thath(xk)G0 for all kG0, 1, 2, . . .. Consider the convex problem

min x1 ,

s.t. h(x1 , x2)G0, g(x1 , x2)⁄0,

where

h(x1 , x2)Gx2 , g(x1 , x2)G[1(x21A1)2C4x2

2Cx21A1]2.

The feasible set of this problem is [−1, 1]B0. The sequence xkG

(−1k, 1k) satisfies the AGP property but converges to the feasible point(0, 0), which is not the minimizer of the problem.

This fact has some algorithmic consequences. On one hand, for manyiterative algorithms, to assume that h(xk)G0 for all k, when h is an affinefunction, is not a serious restriction. In fact, for these methods, the linearconstraints are satisfied naturally at every iteration, so that the sufficiencyproperty holds.

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On the other hand, Theorem 3.1 and the counterexample above seemto indicate that, when we have an algorithm for (1)–(2) with the AGP prop-erty, it is better to formulate the problem using a pair of inequalities replac-ing each equality constraint. This has a practical interpretation: theapproximate projected gradient is associated to an explicit or implicit phaseof many nonlinear programming algorithms, where we try to improve thefunctional value on an approximation of the feasible set. This is called thehorizontal step in many sequential quadratic programming algorithms (Ref.16) and the optimality phase in some inexact-restoration methods (Refs.13–14). The two-inequality reformulation of hj (x)G0 allows one to improvenot only the functional value, but also the feasibility when performing thatphase. In fact, roughly speaking, the domain in which the functional valueis improved at iteration k has, up to second order, the feasibility of hj (x

k)if we use the equality formulation. But using two inequalities, the feasibilityof the new point with respect to hj can be (with second-order error) betweenAhj (x

k) and hj (xk). Even truly feasible points can be included when, in the

approximate region, each equation is described by two inequalities.Of course, there is a strong reason for not using the two-inequality

reformulation: subproblems are much more difficult than when we useequalities. For example, in sequential quadratic programming algorithmswith only equality constraints, the linear algebra of each iteration can bereduced to one or two matrix factorizations, whereas the combinatorialaspect cannot be avoided when using inequalities. See, for example Ref. 16.

This discussion motivates the introduction of a strong AGP conditionin the following section.

4. Strong Approximate Gradient Projection Condition

We define now the strong approximate gradient projection (SAGP)dS (x, γ ). The SAGP vector is the AGP vector related to problem (1)–(2),when each equality constraint hj (x)G0 is transformed into two inequalityconstraints in the obvious way. For all x∈ n, γ ∈[−S, 0],ΩS (x, γ ) isdefined as the set Ω(x, γ ) related to the two-inequality reformulation. Forexample, ΩS (x, −S) is the set of points that satisfy

g′i (x)(zAx)⁄0, if gi (x)¤0,

g(x)Cg′i (x)(zAx)⁄0, if gi (x)⁄0,

and

0⁄hj (x)Ch′j (x)(zAx)⁄hj (x), if hj (x)¤0,

0¤hj (x)Ch′j (x)(zAx)¤hj (x), if hj (x)⁄0.

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Accordingly,

dS (x, γ )GPΩS (x, γ )(xA∇f (x))Ax.

We say that the strong AGP property is fulfilled when, for somesequence,

lim dS (xk, γ )G0.

In the following lemma, we prove that the strong AGP property implies theAGP property.

Lemma 4.1. For all γ ∈[−S, 0], if xk→x* and dS (xk, γ )→0, thend(xk, γ )→0.

Proof. Observe that

Ω(x, γ )⊂ΩS (x, γ ), for allx∈n, γ ∈[−S, 0].

Let us call

wkGPΩS(xk, γ )(x

kA∇f (xk)),

ykGPΩ(xk, γ )(xkA∇f (xk)).

Since xk, yk∈Ω(xk, γ )⊂ΩS (xk, γ ), we have, by the definition of projectionsfor all t∈[0, 1],

xkA∇f (xk)Awk2

⁄ (xkA∇f (xk))A(xkCt(ykAxk)) 2

⁄ t(xkA∇f (xk)Ayk)A(1At)∇f (xk) 2⁄ ∇f (xk) 2.

So,

(12) wkAxk2C(wkAxk)T∇f (xk)

⁄ (t22) ykAxk2Ct(ykAxk)T∇f (xk)⁄0,

for all t∈[0, 1].Taking limits in the above inequalities and using the fact that

wkAxk→0, we obtain that, for all t∈[0, 1],

limk→S

(t22) ykAxk2Ct(ykAxk)T∇f (xk)G0.

Therefore, for all t∈(0, 1],

limk→S

(t2) ykAxk2C(ykAxk)T∇f (xk)G0. (24)

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Since ykAxk is bounded, this implies that

(ykAxk)T∇f (xk)G0.

Therefore, by (24) with tG1, we obtain that ykAxk→0, as we wanted toprove.

Since the SAGP comes from a reformulation of the constraints (5),Theorem 2.1 can be applied and the following corollary also holds.

Corollary 4.1. If x* is a local minimizer of (1)–(2) and γ ∈[−S, 0],there exists a sequence xk⊂n such that lim xkGx* andlim dS (xk, γ )G0.

Moreover, since by Lemma 4.1 dS (xk, γ )→0 implies that d(xk, γ )→0,the following corollary follows in a straightforward way from Theorem 2.2.

Corollary 4.2. Assume that x* is a feasible point. Let γ ∈[−S, 0]. Sup-pose that there exists a sequence xk→x* such that dS (xk, γ )→0. Then, x*is a Fritz–John point of (1)–(2).

We will show that the strong AGP property is a sufficient conditionfor minimizers of convex problems, without the requirement h(xk)G0. Thisproperty is stated in the following theorem. Therefore, the strong AGPproperty is a necessary and sufficient optimality condition for convexproblems.

Theorem 4.1. Assume that, in the nonlinear program (1)–(2), f and gi

are convex, iG1, . . . , p, and h is an affine function. Let γ ∈[−S, 0]. Supposethat x*∈Ω and xk⊂n are such that lim xkGx* and lim dS (xk, γ )G0.Then, x* is a minimizer of (1)–(2).

Proof. As we mentioned above, the strong AGP direction is the AGPdirection for a reformulation of the problem that does not have equalityconstraints. Therefore, the thesis follows from Corollary 4.1.

5. Conclusions

The optimality condition introduced in this paper is oriented to theanalysis of optimization algorithms. On the one hand, there exist good non-linear programming methods satisfy the new condition. This is a questionof simple verification in some cases. The condition is proved explicitly in

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inexact restoration algorithms (Refs. 13–14) and it can be easily verified inaugmented Lagrangian methods. In other cases (Refs. 17–18), it coulddemand some detailed research analysis.

We conjecture that all reliable algorithms for nonlinear programmingsatisfy stronger theoretical properties than the one that says that feasibleaccumulation points are Fritz–John points. The set of AGP points is asharper approximation to the set of local minimizers than the set of Fritz–John points; perhaps, there can be identified other sharp approximations tothe minimizers that can be linked to the convergence of good minimizationalgorithms. This will be the subject of future research.

The mathematical programming problem with equilibrium constraints(MPEC) is a good example of a problem where the new condition is useful.In the usual formulation, the constraints take the form xigiG0, xi¤0, gi¤

0, which implies that all the feasible points are Fritz–John points. However,it can be shown that few points satisfy the AGP condition. See Ref. 19 foran analysis of the use of nonlinear algorithms for solving MPEC.

Further research is necessary in order to extend the new optimalitycondition to nonsmooth optimization, variational inequality problems,bilevel programming, and vector optimization (see Refs. 20–25). As in thesmooth case analyzed in this paper, the guiding line comes from having inmind what efficient algorithms do for these problems, what kind of conver-gence results have already been proved for them, and which good and obvi-ous practical behavior has been lost in those results.

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