MINIMAX PROGRAMMING

MINIMAX PROGRAMMING

UNDER NEW INVEXITY ASSUMPTIONS

MARIA VIORICA S¸TEF ˘ANESCU and ANTON S¸TEF ˘ANESCU

We consider the minimax programming problem and focus on optimality condi- tions and duality properties. Some of previously known theoretical results are now restated in terms of (Φ, ρ)-invexity and related properties.

AMS 2000 Subject Classification: 90C26, 90C46.

Key words: minimax programming, invexity, duality.

1. INTRODUCTION

The theory of mathematical programming has grown remarkably after generalized convexity has been used in the settings of optimality conditions and duality theory. In 1981, Hanson [4] showed that both weak duality and Kuhn- Tucker sufficiency for optimum hold when convexity is replaced by a weaker condition. This condition, called invexity by Craven [2], was further studied for more general problems and was a source of a vast literature. After the works of Hanson and Craven, other types of differentiable functions have been introduced with the intent of generalizing invex functions from different points of view. Hanson and Mond [5] introduced the concept of F-convexity and Jeyakumar [3] generalized Vial’s ρ-convexity ([7]) by introducing the concept ofρ-invexity. The concept of generalized (F, ρ)-convexity, introduced by Preda [6], is in turn an extension of the above properties and was used by several authors to obtain relevant results.

(F, ρ)-convexity was recently generalized to (Φ, ρ)-invexity by Caristi et al. [1] to extend the main theoretical results of mathematical programming.

Now we consider the minimax programming problem involving (Φ, ρ)- invex functions, and obtain optimality conditions and duality results.

2. (Φ, ρ)-INVEXITY AND RELATED EXTENSIONS Letϕ:X₀ →Rbe a differentiable function (X₀ ⊆Rⁿ), anda∈X₀. In the definitions below, an element of the (n+ 1)-dimensional Euclidean space Rⁿ⁺¹ is represented as an ordered pair (y, r), with y ∈ Rⁿ and r ∈ R,

REV. ROUMAINE MATH. PURES APPL.,52(2007),3, 367–376

and Φ is a real-valued function defined onX₀×X₀×Rⁿ⁺¹such that Φ(x, a, .) is convex onRⁿ⁺¹ and Φ(x, a,(0, r))≥0 for every (x, a)∈X₀×X₀ andr ∈R₊. Now, fix a nonvoid subsetX of X₀,a point a∈X₀, and a real number ρ.Following [1] we recall some basic notions and define some new extensions of them.

Definition 1. ϕis said to be (Φ, ρ)-invexat awith respect to X if (1) ϕ(x)−ϕ(a)≥Φ(x, a,(∇ϕ(a), ρ)), ∀x∈X.

Remark. For Φ(x, a,(y, ρ)) =F(x, a, y) +ρd²(x, a),withF sub-linear in the third argument, (Φ, ρ)-invexity reduces to Preda’s (F, ρ)-convexity. More- over, an example in [1] shows that (Φ, ρ)-invexity may hold when other known invexity-type conditions are not satisfied.

Definition 2. ϕis said to bepseudo (Φ, ρ)-invex at awith respect to X ifϕ(x)−ϕ(a)≥0 whenever Φ(x, a,(∇ϕ(a), ρ))≥0 for somex∈X.

Definition 3. ϕis said to bequasi(Φ, ρ)-invexat awith respect toX if Φ(x, a,(∇ϕ(a), ρ))≤0,whenever ϕ(x)−ϕ(a)≤0 for some x∈X.

Definition 4. ϕ is said to be semistrict quasi (Φ, ρ)-invex at a with respect to X if Φ(x, a,(∇ϕ(a), ρ)) < 0, whenever ϕ(x)−ϕ(a) < 0 for some x∈X.

ϕ is said to be (Φ, ρ)-invex (pseudo (Φ, ρ)-invex, quasi (Φ, ρ)-invex, semistrict quasi (Φ, ρ)-invex) on X₀ if it is (Φ, ρ)-invex (pseudo (Φ, ρ)-invex, strictly pseudo (Φ, ρ)-invex, quasi (Φ, ρ)-invex, semistrict quasi (Φ, ρ)-invex) at afor everya∈X₀.

Obviously, (Φ, ρ)-invexity implies pseudo,quasi and semistrict quasi (Φ, ρ)- invexity.

3. MINIMAX PROGRAMMING

The problem to be considered here is the minimax programming problem (M) : min max

1≤i≤p f_i(x), g(x)≤0, x∈X₀,

where X₀ is a nonvoid open subset of Rⁿ, f_i, i ∈ {1,2, . . . , p}, and g_j, j ∈ {1,2, . . . , m}, are differentiable real-valued functions defined on X₀; f and g are the vector functions f = (f₁, . . . , f_p) : X₀ → R^p, g = (g₁, g₂, . . . , g_m) : X₀ →R^m,and the symbol “≤” stands both for the usual inequality inRand for the component-wise inequality in a multi-dimensional Euclidean space

LetX be the set of all feasible solutions X ={x∈X₀ |g(x)≤0}

of problem (M). The problem (M₁) below is “equivalent” to (M):

(M₁) : minν, g(x)≤0, f(x)≤νe, x∈X₀. Hereeis the p-dimensional vector with all entries 1.

The equivalence means that if x is a feasible (optimal) solution of (M), then (x, ν),whereν = max

1≤i≤p f_i(x),is a feasible (optimal) solution of (M₁) and, conversely, if (x, ν) is a feasible (optimal) solution of (M₁) then xis a feasible (optimal) solution of (M).

Note that (M₁) is a smooth problem while (M) is not, and then we will appeal to (M₁) as a substitute of (M) when differentiability is a necessary condition.

Another restatement of our minimax programming problem is (M₂) : minF(x), x∈X₀,

where ∆ is the standard (p −1)-dimensional simplex of R^p, ∆ = {µ = (µ₁, . . . , µ_p) |µ_i≥0, i= 1, . . . , p,_p

i=1µ_i = 1}, and F(x) = sup

(µ,λ)∈∆×R^m₊ L(x, µ, λ), withL(x, µ, λ) =_p

i=1µ_if_i(x) +_m

j=1λ_jg_j(x).

It is obvious that if X =∅, then (M) and (M₂) have the same optimal solutions. In fact, if X = ∅, then O(M) = O(M₁) = O(M₂). (Here, O(P) is the generic notation for the set of all optimal solutions of problem (P).)

In the next two sections we will show that some fundamental results con- cerning optimality and duality for the minimax programming can be restated in terms of (Φ, ρ)-invexity and related properties.

For the sake of concision we will use the shorthands “(Φ, ρ)-invexity (pseudo-, quasi-, etc.) at a” to mean “(Φ, ρ)-invexity (pseudo-, quasi-, etc.) at a, with respect to X”. When (Φ, ρ)- invexity is defined with respect to another set, this will be specified.

4. OPTIMALITY CONDITIONS

In this section we extends some necessary and sufficient conditions for optimality under various (Φ, ρ)-invexity conditions defined above.

Since (M₁) is a smooth scalar optimization problem, we can follow the usual procedure involving the associated Lagrange function in order to obtain the Kuhn-Tucker conditions.

Definition5. ((a, ν), µ, λ)∈X×R×R^p₊×R^m₊ is said to be aKuhn-Tucker point of(M₁) if

(2)

p i=1

µ_i∇f_i(a) + m j=1

λ_j∇g_j(a) = 0,

(3)

m j=1

λ_jg_j(a) = 0,

(4) f(a)≤νe,

(5) µ, f(a)=ν,

(6) e, µ= 1.

Note that ifJ(a) stands for the set of indices of all active restrictions at a∈X, J(a) ={j |g_j(a) = 0},then (2) and (3) become, respectively,

(7)

p i=1

µ_i∇f_i(a) +

j∈J(a)

λ_j∇g_j(a) = 0,

(8)

j∈J(a)

λ_jg_j(a) = 0.

In what follows we will use both versions of these conditions.

SetM(a) =

i|f_i(a) = max

1≤≤p f(a) .

Definition6. A triple (a, µ, λ) ∈X×R^p₊×R^m₊ is said to be aKuhn-Tucker point of(M) if it verifies (3) and the conditions

i∈M(a)

µ_i∇f_i(a) +

j∈J(a)

λ_j∇g_j(a) = 0,

(9)

i∈M(a)

µ_i= 1.

First, we will prove that the Kuhn-Tucker conditions are satisfied by optimal solutions of (M) (or (M₁)) if these problems also satisfy some (Φ, ρ)- invexity conditions.

Theorem 1. Let (a, ν) be an optimal solution of (M₁). Assume that Slater’s constraint qualification is satisfied for all restrictions indexed inJ(a).

Assume that for eachj∈J(a)the function g_j is semistrict quasi (Φ, ρ_j)-invex ata for some ρ_j ≥0.Then there exist µ∈R^p₊ and λ∈R^m₊ such that (2)–(6) are verified by ((a, ν), µ, λ).

Proof. Since the problem (M₁) is smooth, the Fritz-John necessary con- ditions claim the existence of non-negative multipliersw∈R, u∈R^p,v∈R^m such that

(10) w− e, u= 0, p

i=1

u_i∇f_i(a) + m j=1

v_j∇g_j(a) = 0, f(a)≤νe, u, f(a)−νe= 0, v, g(a)= 0,

(11) w+

p i=1

u_i+ m j=1

v_j >0.

We have to prove thatw+_p

i=1u_i >0.

Otherwise, if w= 0 and u= 0,it follows from (11) thatv₀=_m

j=1v_j >

0.Settingv_j =v_j/v₀, j = 1,2, . . . , m,equation (7) becomes

j∈J(a)v_j∇g_j(a) = 0.It then follows from the properties of Φ that

0≤Φ

x, a,

j∈J(a)

v_j∇g_j(a),

j∈J(a)

v_jρ_j

≤

j∈J(a)

v_jΦ(x, a,(∇g_j(a), ρ_j)) for everyx∈X.

Let x^∗ ∈ X satisfy Slater’s conditions g_j(x^∗) < 0, ∀j ∈ J(a). Then, because eachg_j is semistrict quasi (Φ, ρ_j)-invex, we have

j∈J(a)

v_jΦ(x^∗, a,(∇g_j(a), ρ_j))<0, so that we have reached a contradiction.

Now, observe that the above inequality together (10) say thate, µ= 1, where µ_i = u_i/w. Finally, set λ_j = _w¹v_j, j = 1, . . . , m, and the proof is complete.

Theorem2. Let a be an optimal solution of (M). Assume that Slater’s constraint qualification is satisfied for all restrictions indexed inJ(a).Assume that for eachj∈J(a) the functiong_j is semistrict quasi(Φ, ρ_j)-invex atafor someρ_j ≥0.Then there exist µ∈R^p₊ and λ∈R^m₊ such that (2), (3)and (9) are verified by (a, µ, λ).

Proof. Since a∈O(M) we have (a, ν) ∈O(M₁),where ν = max

1≤i≤p f_i(a).

Then, by Theorem 1, there exist µ∈ R^p₊ and λ∈ R^m₊ such that ((a, ν), µ, λ) is a Kuhn-Tucker point of (M₁). The definition of ν makes (4) trivial, and it follows from (5) thatµ_i= 0 if i /∈M(a).Hence (6) reduces to (9).

The next two results concern the sufficiency of Kuhn-Tucker conditions when (Φ, ρ)-invexity is assumed

Theorem3. Let ((a, ν), µ, λ) be a Kuhn-Tucker point of (M₁).Assume for each j ∈ J(a), that _p

i=1µ_if_i is pseudo (Φ, ρ₀)-invex at a, g_j is quasi (Φ, ρ_j)-invex at a, and ρ₀+

j∈J(a)λ_jρ_j ≥ 0. Then a ∈ O(M) and (a, ν) ∈ O(M₁).

Proof. By (4), (a, ν) is a feasible solution of (M₁). Let (x, ν) be an arbitrary feasible solution of (M₁). Two situations should be considered:

(i)λ= 0; (ii) λ= 0.

In case (i), we have

(12) Φ

x, a,

^p

i=1

µ_i∇f_i(a), ρ₀

≥0.

If case (ii) holds, letw= 1 +

j∈J(a)λ_j,so that we have 1/wΦ

x, a,

^p

i=1

µ_i∇f_i(a), ρ₀

+

j∈J(a)

λ_j/wΦ(x, a,(∇g_j(a), ρ_j))≥

≥Φ

x, a,

1/w p i=1

µ_i∇f_i(a) +

j∈J(a)

λ_j/w∇g_j(a), ρ₀/w+

j∈J(a)

λ_jρ_j/w

≥0.

Since the g_j are quasi (Φ, ρ_j)-invex at a and g_j(x)−g_j(a) =g_j(x) ≤ 0 for eachj ∈ J(a),we have

j∈J(a)λ_j/wΦ(x, a,(∇g_j(a), ρ_j))≤0, so that we arrive again to (12).

Now, by (12), (Φ, ρ₀)-invexity implies the inequality p

i=1

µ_if_i(x)− p i=1

µ_if_i(a)≥0.

But f_i(x) ≤ν, _p

i=1µ_if_i(a) = ν and _p

i=1µ_i = 1. Therefore, the inequality above implies the inequalityν−ν≥0.

Theorem4. Let (a, µ, λ) be a Kuhn-Tucker point of (M). Assume that f_i is semistrict quasi (Φ, ρ_0i)-invex at afor each i∈M(a) g_j is quasi (Φ, ρ_j)- invex at a for each j ∈ J(a), and

i∈M(a)µ_iρ_0i+

j∈J(a)λ_jρ_j ≥ 0. Then, a∈ O(M).

Proof. By Definition 5,ais a feasible solution of (M). Putν = max

1≤i≤p f_i(a).

Thus,f_i(a) =νfor everyi∈M(a).Suppose thatais not optimal. Then there existsx ∈X such that ν = max

1≤i≤p f_i(x) < ν. Since (a, µ, λ) is a Kuhn-Tucker point, we have

i∈M(a)

u_i∇f_i(a) +

j∈J(a)

v_j∇g_j(a) = 0,

whereu_i =µ_i/µ₀, i∈M(a),v_j =λ_j/µ₀,j∈J(a), µ₀ = 1 +

j∈J(a)λ_j.Also,

i∈M(a)u_iρ_0i+

j∈J(a)v_jρ_j ≥0, and then Φ

x, a,

i∈M(a)

u_i∇f_i(a) +

j∈J(a)

v_j∇g_j(a), p i=1

u_iρ_0i+

j∈J(a)

v_jρ_j

≥0.

Moreover,

i∈M(a)

u_iΦ(x, a,(∇f_i(a), ρ_0i)) +

j∈J(a)

v_jΦ(x, a,(∇g_j(a), ρ_j))≥0 hence

i∈M(a)

u_iΦ(x, a,(∇f_i(a), ρ_0i))≥0.

Fori∈M(a) we havef_i(x)−f_i(a)≤ν−ν <0,so that Φ(x, a,(∇f_i(a), ρ_0i))<

0.Sinceu∈R^p₊and_p

i=1u_i >0,we have

i∈M(a)u_iΦ(x, a,(∇f_i(a), ρ_0i))<0, contradicting the previous inequality.

5. DUALITY

In strict formal terms, we will consider here two couples of dual problems defining the Wolfe dual of (M₁) and the generalized dual of (M₂). Due to the equivalences pointed out in Section 3, both problems are duals of (M).

As we have already observed, problem (M₁) is smooth, so that it admits a Wolfe dual

(WM) :

max p

i=1

µ_if_i(y) + m j=1

λ_jg_j(y)

, p

i=1

µ_i∇f_i(y) + m j=1

λ_j∇g_j(y) = 0, y∈X₀, µ∈∆, λ∈R^m₊. The equivalence of (M) and (M₁) allow us to refer to (WM) as to the dual of (M).In fact, the properties listed bellow actually relate by duality the two problems.

Theorem5. Letxbe a feasible solution of(M)and(y, µ, λ)a feasible so- lution of(WM).Assume thatf_i is(Φ, ρ_0i)-invex at yfor eachi∈ {1,2, . . . , p}, g_j is(Φ, ρ_j)-invex atyfor eachj∈ {1,2, . . . , m},and_p

i=1µ_iρ_0i+_m

j=1λ_jρ_j ≥ 0.Then _p

i=1µ_if_i(y) +_m

j=1λ_jg_j(y)≤ max

1≤i≤p f_i(x).

Proof. Settingw= 1 +_m

j=1λ_j, u_i =µ_i/w, and u_j =λ_j/w, we have Φ

x, y,

^p

i=1

u_i∇f_i(y) + m j=1

v_j∇g_j(y), p i=1

u_iρ_0i+ m j=1

v_jρ_j

≥0,

hence p i=1

u_iΦ(x, y,(∇f_i(y), ρ_0i)) + m j=1

v_jΦ(x, y,(∇g_j(y), ρ_j))≥0.

Furthermore, the invexity off_i and g_j implies p

i=1

u_i(f_i(x)−f_i(y)) + m j=1

v_j(g_j(x)−g_j(y))≥0.

Multiplying by w,the last inequality yields p

i=1

µ_if_i(y) + m j=1

λ_jg_j(y)≤ p i=1

µ_if_i(x) + m j=1

λ_jg_j(x).

Since x∈X, it follows that p

i=1

µ_if_i(x) + m j=1

λ_jg_j(x)≤ p i=1

µ_if_i(x)≤ max

1≤i≤pf_i(x).

Remark. Under invexity assumptions, if (x, ν) is a feasible solution of (M₁), then _p

i=1µ_if_i(y) +_m

j=1λ_jg_j(y)≤ν.

Theorem6. Let a be an optimal solution of (M). Assume that Slater’s constraints qualification is satisfied for all restrictions indexed in J(a). If, f_i is (Φ, ρ_0i)-invex on X₀, for each for each i ∈ {1,2, . . . , p}, g_j is (Φ, ρ_j)- invex on X₀, for each j ∈ {1,2, . . . , m}, where ρ_0i, ρ_j ≥ 0, then there exist µ and λsuch that (a, µ, λ) is an optimal solution of (WM) and _p

i=1µ_if_i(a) + _m

j=1λ_jg_j(a) = max

1≤i≤p f_i(a)

Proof. By Theorem 2 there existµ∈R^p₊,andλ∈R^m₊ such that (a, µ, λ) is a Kuhn-Tucker point of (M). Then, (a, µ, λ) is a feasible solution of (WM) and since_m

j=1λ_jg_j(a) = 0,we have p

i=1

µ_if_i(a) + m j=1

λ_jg_j(a) =

i∈M(a)

µ_if_i(a) = max

1≤i≤p f_i(a) Then by the optimality of (a, µ, λ) follows by Theorem 5.

Let us now generalize duality with respect to (M₂).

The generalized dual of (M) is the scalar optimization problem (GM) : maxG(µ, λ), (µ, λ)∈∆×R^m₊,

whereG(µ, λ) = inf

x∈X0

L(x, µ, λ).

The weak duality property follows, with no restriction, from the universal minimax inequality

sup

(µ,λ)∈∆×R^m₊ inf

x∈X0

L(x, µ, λ)≤ inf

x∈X0

sup

(µ,λ)∈∆×R^m₊ L(x, µ, λ).

Thus,

(13) sup

(µ,λ)∈∆×R^m₊ G(µ, λ)≤ inf

x∈X0 F(x).

For a direct duality theorem, (Φ, ρ)-invexity is sufficient.

Theorem 7. Let a ∈ X be an optimum solution of (M₂). Assume that Slater’s constraints qualification is satisfied for all restrictions indexed inJ(a).

If f_i is (Φ, ρ_0i)-invex at a with respect to X₀ for each i ∈ M(a) and g_j is (Φ, ρ_j)-invex at a with respect to X₀, for each j ∈ J(a) where ρ_0i, ρ_j ≥ 0, then there exist µ and λ such that (µ, λ) is an optimal solution of (GM) and G(µ, λ) =F(a).

Proof. SinceO(M) =O(M₂), ais an optimal solution of (M) andF(a) =

1≤i≤pmax f_i(a).Then, by Theorem 2, these exist µ_i ∈R^p₊ and λ∈R^m₊ such that

i∈M(a)

µ_i∇f_i(a) +

j∈J(a)

λ_j∇g_j(a) = 0,

j∈J(a)

λ_jg_j(a) = 0, and

i∈M(a)µ_i= 1.

Forx∈X₀ we have

i∈M(a)

u_iΦ(x, a,(∇f_i(a), ρ_0i)) +

j∈J(a)

v_jΦ(x, a,(∇g_j(a), ρ_j))≥0, where,u_i=µ_i/w, v_j =λ_j/w, w= 1 +_m

j=1λ_j. Now, use the invexity assumptions to obtain

i∈M(a)

u_i(f_i(x)−f_i(a)) +

j∈J(a)

v_j(g_j(x)−g_j(a))≥0.

Let µ∈R^p₊ withµ_i = µ_i ifi∈M(a) andµ_i = 0 otherwise. Obviously, µ∈∆ and the last inequality is equivalent to

p i=1

µ_if_i(x) + m j=1

λ_jg_j(x)≥ p

i=1

µ_if_i(a).

Since _p

i=1µ_if_i(a) =

i∈M(a)µ_if_i(a) = max

1≤i≤p f_i(a) = F(a), and the above inequality holds for every x∈X₀,we have

x∈Xinf0

p i=1

µ_if_i(x) + m j=1

λ_jg_j(x)

≥F(a),

i.e. G(µ, λ)≥F(a).Then (13) implies the optimality of (µ, λ) and the equation G(µ, λ) =F(a).

Remark. The last equation is just the minimax equation

x∈Xmin0 sup

(µ,λ)∈∆×R^m₊ L(x, µ, λ) = max

(µ,λ)∈∆×R^m₊ inf

x∈X0 L(x, µ, λ), so that the above theorem is a minimax theorem forL.

Acknowledgements. This research was partly supported by CNCSIS Grant 27694/2005.

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Received May 2, 2006 Academy of Economic Studies Department of Mathematics

Calea Dorobantilor 15–17, Bucharest, Romania and

University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania