Necessary conditions and duality for inexact nonlinear semi-infinite programming problems

First order necessary conditions and duality results for general inexact nonlinear programming problems formulated in nonreflexive spaces are obtained. The Dubovitskii–Milyutin approach is the main tool used. Particular cases of linear and convex programs are also analyzed and some comments about a comparison of the obtained results with those existing in the literature are given. 1 Problem statements In this paper we deal with the following inexact nonlinear programming problem: min x J (x) = min x max c∈C f (x, c), s.t. h(x, ε) ∈ B, a.e. ε ∈ [0, 1] , g(x, ε) ∈ D + R+, ε ∈ [0, 1] , (1) with the assumptions: (H1) x ∈ Rn and C is a convex and compact set in Rm, (H2) The functions f : Rn × Rm −→ R, h : Rn × [0, 1] −→ Rp and g : R n × [0, 1] −→ Rq are supposed to be continuously differentiable with respect to x, (H3) f and g are continuous and h is measurable and essentially bounded with respect to their second arguments, J. A. Gómez Departamento Ingeniería Matemática, Universidad de La Frontera, Temuco, Chile E-mail: [email protected] P. Bosch (B) Facultad de Ingeniería, Universidad Diego Portales, Santiago, Chile E-mail: [email protected] 46 J. A. Gómez and P. Bosch (H4) The sets B ⊂ Rp and D ⊂ Rq are supposed to be convex with nonempty interior. Some examples of important particular cases of (1) are the inexact semi-infinite NLP problem: min x max c∈C f (x, c) s.t. g(x, ε) ∈ D + R+, ∀ε ∈ [0, 1], x ∈ Rn, C ⊂ Rn,D ⊂ Rm, the inexact second order cone LP problem: min x max c∈C cT x s.t. Ax ∈ D + Lm+1, where Lm+1 is the Lorentz cone: L m+1 = (ξ1, . . . , ξm, ξm+1) ∈ Rm+1 : ‖(ξ1, . . . , ξm)‖ ≤ ξm+1 } , and the inexact semi-infinite, semi-definite (LP) NLP problem: min x max c∈C f (x, c) F(x, ε) ∈ B + S+, ∀ε ∈ [0, 1], x ∈ Rn, C ⊂ Rn,B ⊂ Sm, where F(x, ε) = F(x, F1(ε), . . . , Fn(ε)) is a (linear) function on x for n given fixed matrices F1, . . . , Fn belonging to the set Sm of symmetric m × m matrices. B is a convex subset of the cone S+ of positive semi-definite symmetric matrices. All the above problems become exact (i.e. with exact data) when B, C and D are singletons. As first pointed out in Tichatschke et al. (1989) and also in Amaya and Gómez (2001), Vandenberghe and Boyd (1998), all these problems are essentially nonlinear problems, even if f, g are affine, then nonlinear programming tools should be used to obtain necessary optimality conditions (mostly in duality theorem forms) and/or to design computational algorithms. An increasing number of papers and books were published in recent years dealing with duality results and algorithmic ideas for the above exact or inexact problems. As an incomplete list of references on these subjects let us only mention Borwein (1981), Goberna and López (1998), Hettich (1986), Hettich and Kortanek (1993), Ramana et al. (1997), Reemtsen (1991), Reemtsen and Görner (1998), Shapiro (1998), Tichatschke et al. (1989), Vandenberghe and Boyd (1998) for semiinfinite programming, Amaya and de Ghellinck (1997), Amaya and Gómez (2001), Matloka (1992), Soyster (1973, 1974) dealing with inexact LP, and Alizadeh and Goldfarb (2002), Ben Tal and Nemirovski (1998, 2001), Goldfarb (2002), Krishnan and Mitchell (2002, 2003a,b), Nesterov and Nemirovskii (1993), Vandenberghe and Boyd (1996) concerning on theory and methods for semidefinite and second order cone programming. It is well known that those models have a large field of applications, including control of robots, eigenvalue computations, mechanical stress of materials, statistical design and others (see, for instance, Ben Tal and Nemirovski 2001; Lobo et al. 1987; Reemtsen 1991). Furthermore, some practical applications of the above more complex models are recently appearing in the context of robust Inexact nonlinear semi-infinite programming problems 47 optimization models in economy (see, for example, Leibfritz and Maruhn 2005). But in fact, any robust formulation of semi-infinite programming problems would also be considered inexact (see Ben Tal and Nemirovski 2001). The use of duality relations for algorithmic design is a classical idea and this approach generally leads to discretization methods and/or cutting plane algorithms (see, for instance, Goberna and López 1998; Hettich 1986; Hettich and Kortanek 1993; Reemtsen 1991). Necessary optimality conditions in the form of strong duality theorems can be obtained using different approaches, where the more general ones are the parameter perturbation and conjugate functionals approach (see, for instance, Rockafellar 1974; Shapiro 1998, 2001), the Lagrangian functional and the Dubovitskii–Milyutin theory (see, for instance, Amaya and Gómez 2001). The Dubovitskii–Milyutin (D–M) approach, first published in Dubovitskii and Milyutin (1965), became very popular and is described extensively in the book of Girsanov (1972). It was used by the authors to obtain a general version of Karush– Kuhn–Tucker theorem for problems containing equality and inequality operator constraints (Bosch and Gómez 1997), the local Pontryagin’s maximum principle for smooth optimal control problems with mixed state constraints (Bosch and Gómez 2000), and more recently, in Amaya and Gómez (2001) and Gómez et al. (2005), to obtain appropriate definitions of dual problems and strong duality theorems for inexact linear programming and inexact semi-infinite linear programming. An extense literature about duality can be found in the references. Our main objective is to obtain first order necessary conditions for problem (1) and strong duality relations in nonlinear and convex cases. For this end we shall use (D–M) approach defining appropriate cones of decreasing, feasible and tangent directions and computing the respective dual cones. Then, we shall apply the following well known (D–M) Lemma: Lemma 1 Let K1, . . . , K p, K p+1 be convex cones with apex θ in a topological vector space X (θ represents the origin of X), where K1, . . . , K p are open. We suppose that we are not in the singular case, where one cone is empty and all the others are the whole space X. Then, ⋂p+1 i=1 Ki = ∅ if and only if there exist linear functionals fi ∈ K ∗ i , i = 1, . . . , p + 1, not all zero, such that: f1 + · · · + f p + f p+1 = θ∗, where symbol K ∗ i denotes the dual cone of the set Ki , i.e. the set of all linear and continuous functionals which are non negative on Ki . The direct application of (D–M) approach to problem (1) in the space Rn is not clear. It is well known that, even for the linear case, infinite dimensional spaces appear naturally to obtain necessary conditions and duality results for exact SIP problems, for example, the space of generalized finite sequences (Goberna and López 1998, Chap. 2) or the space of continuous functions (Shapiro 1998, Cap. 4). Incidentally, any generalization of these results to the inexact case must consider the non-reflexive space of continuous functions Cq for the inequality constraints. These are the reasons explaining why infinite dimensional and non reflexive spaces were considered here. 48 J. A. Gómez and P. Bosch Problem (1) shall be transformed in the following way. Define the functional sets: B := {y ∈ L∞ : y(ε) ∈ B, a.e. ε ∈ [0, 1]} , D := z ∈ Cq : z(ε) ∈ D, ∀ε ∈ [0, 1] } , P := z ∈ Cq : z(ε) ≥ 0, ∀ε ∈ [0, 1] } , and the constraint operators: H : Rn × L∞ −→ L∞ and G : Rn × Cq −→ Cq : H(x, y)(ε) = h(x, ε)− y(ε), G(x, z)(ε) = g(x, ε)− z(ε). Here L∞ = L∞ ([0, 1] ,Rp) denotes the Banach space of essentially bounded measurable Rp-valued functions and Cq = C ([0, 1] ,Rq) is the Banach space of continuousRq -valued functions, with the usual (essential) supremum norm defined on each space. By assumptions, operators H and G become continuously Frechet differentiable with respect to (x, y) ∈ Rn ×L∞ and (x, z) ∈ Rn ×Cq respectively, and the functional J is directional differentiable. We shall denote Frechet differentials with symbol D and directional differentials with symbol δ. Note that the sets B and D have nonempty interior due to the same property of B and D. This last observation is another reason to consider non-reflexive spaces L∞ and Cp, since for the usual reflexive spaces L p (1 < p < ∞) the sets B and D does not have interior points. The constraints shall be written as follows: H(x, y) = θ∞, (2) G(x, z) θC ⇐⇒ G(x, z) ∈ P, (3) where θC (θ∞) denotes the null function on Cq (L∞). In fact, all the results in this paper are still true if we suppose P a convex cone in Cq which is pointed (θC ∈ P), proper (P ∩ {−P} = θC ) and with nonempty interior (int(P) = ∅). In this more general case we keep the notation G(x, z) θC for G(x, z) ∈ P, since P defines a partial order in Cq , and we read the relation G(x, z) θC as G(x, z) ∈ int (P). In addition, convexity assumption implies P = cl(int(P)). Problem (1) becomes: min x J (x) = min x max c∈C f (x, c), s.t. H(x, y) = θ∞ G(x, z) θC (x, z, y) ∈ Rn × D × B. (4) The hypothesis (H3) about function ε → h(x, ε) demands the new variable y(.) ∈ L∞, and this make the first constraint of (4) equivalent to the corresponding one in (1). The continuity assumption for the variable z(.) should be analized in order to problems (1) and (4) be considered equivalent. In fact, the first component of any solution (x, z, y) of (4) produces, trivially, a solution x for (1). On the other hand, if x is a solution of (1), g(x, ε) ∈ D + R+, ∀ε ∈ [0, 1]. Then, we must prove that, for each x , there exists a continuous function z(.) such that z(ε) ∈ D and gi (x, ε) ≥ zi (ε), i = 1,m. This statement is equivalent to the relation z(ε) ∈ x (ε) = (g(x, ε)−R+)∩D, and the made assumptions give that every set x (ε) Inexact nonlinear semi-infinite programming problems 49 is convex. Therefore, the theorem of continuous selection (see Michael 1956–1957 or Michael 1970) guarantees the existence of a continuous function z(.) with the necessary properties. Finally, defining y(ε) = h(x, ε), we have that (x, z, y) is also a solution of (4) and both problems become equivalent. Main parts of the following proof are extensions of the proof of the linear case (see Gómez et al. 2005). Let’s suppose (x̄, z̄, ȳ) ∈ X = Rn ×Cq × L∞ is a feasible solution of problem (4). We define the cone: K0 = { (d, k, r) ∈ X | max c∈C0 ∇x f (x̄, c)(d) < 0 } , of decreasing directions of J , where C0 denotes the compact set: C0 = C0(x̄) := { c̄ ∈ C : max c∈C f (x̄, c) = f (x̄, c̄) } , and we also introduce the corresponding cones: K H = {(d, k, r) ∈ X | DH(x̄, ȳ)(d, r) = θ∞} , of tangent directions for the equality constraint, KG = {(d, k, r) ∈ X | ∃τ0 > 0 : G(x̄, z̄)+ τ DG(x̄, z̄)(d, k) θC , ∀τ ∈ (0, τ0)} , of feasible directions for the inequality constraint, and KB = {(d, k, r) ∈ X | r = λr (y − ȳ), λr > 0, y ∈ int(B)} , KD := {(d, k, r) ∈ X | k = λk(z − z̄), λk > 0, z ∈ int(D)} of feasible directions for the set constraints given by B and D. It’s easy to see that all this sets are convex cones with apex at (0n, θC , θ∞) ∈ X. They are open sets with the exception of K H which is closed. 2 Computation of dual cones The description of the dual cones of KB and KD is given by the support functionals of B and D at the points ȳ and z̄ respectively. This result is well known and can be found, for example, in Girsanov (1972): K ∗ B = { (0t, θ∗ C , φ) ∈ X∗ | φ(ȳ) ≤ φ(y),∀y ∈ B } , K ∗ D = { (0t, φ, θ∗ ∞) ∈ X∗ | φ(z̄) ≤ φ(z),∀z ∈ D } . Here X∗ = (Rn)t × C∗ q × L∗∞ is the topological dual space of X and θ∗ ∞ and θ∗ C are the zero functionals of the dual spaces L∗∞ and C∗ q respectively. For the computation of K ∗ G we proceed in several steps, proving the following statements: 50 J. A. Gómez and P. Bosch (1) If KG = ∅ then its closure cl(KG) is the cone K̃G given by: K̃G = {(d, k, r)∈ X | ∃τ0 > 0 : G(x̄, z̄)+ τ DG(x̄, z̄)(d, k) θC ,∀τ ∈ (0, τ0)} , (2) The cone RG , with vertex in (0n,G(x̄, z̄), θ∞) ∈ X (G(x̄, z̄) θC ), is contained in cl(KG), where: RG = {(d, k, r) ∈ X | G(x̄, z̄)+ DG(x̄, z̄)(d, k) θC } , (3) If RG = ∅ and if G(x̄, z̄) θC but G(x̄, z̄) θC , i.e. G(x̄, z̄) ∈ P\int(P) then, the dual cone of RG is given by: R∗ G = { f ∈ X∗ | f =(q∗ ◦ DG(x̄, z̄), θ∗ ∞), q∗ ∈C∗ q , q∗ ≥ 0, q∗ (G(x̄, z̄))=0 } . By well known results in Girsanov (1972) and since RG ⊂ cl(KG) from (1)–(3) we have: K ∗ G = (cl(KG))∗ ⊂ R∗ G or K ∗ G ⊂ { f ∈ X∗ | f = (q∗ ◦ Dx G(x̄, z̄), q∗ ◦ DzG(x̄, z̄), θ∗ ∞), q∗ ∈ P∗, q∗ × (G(x̄, z̄)) = 0 . Remark 2 Therefore, we do not compute the dual cone of KG but we will obtain a characterization of the elements of K ∗ G without its exact calculation. Note that since (x̄, z̄, ȳ) is feasible, we always have G(x̄, z̄) θC . If G(x̄, z̄) θC the inequality constraint can be ignored if we are looking for first order local optimality conditions, since by convexity KG = X . Then, the assumption G(x̄, z̄) ∈ P\int(P) is in fact the interesting case. On the other hand, RG = ∅ is an important hypothesis and it will be considered as part of the constraint qualification conditions that we shall introduce in the next section. Lemma 3 If KG = ∅ then cl(KG) = K̃G . Proof Trivially cl(KG) ⊂ K̃G . For the other inclusion, if (d, k, r) ∈ K̃G and (d1, k1, r1) ∈ KG , then by convexity, the sequence: (dn, kn, rn) = 1 n (d1, k1, r1)+ ( 1 − 1 n ) (d, k, r) ∈ KG for all n ∈ N, and this means (d, k, r) ∈ cl(KG). Lemma 4 If G(x̄, z̄) θC then RG ⊂ int(KG) ⊂ cl(KG). Proof Let be (d, k, r) ∈ RG ⇔ DG(x̄, z̄)(d, k) ∈ −G(x̄, z̄) + int(P). Since G(x̄, z̄) ∈ P we have θC ∈ −G(x̄, z̄)+ P , but P is a convex cone with nonempty interior and therefore −G(x̄, z̄)+ P is also a convex cone with nonempty interior. Then, we can write: (1 − τ)θC + τ DG(x̄, z̄)(d, k) ∈ −G(x̄, z̄)+ int(P), ∀τ ∈ (0, 1) , and this means that (d, k, r) ∈ int(KG) ⊂ cl(int(KG)) = cl(KG). Inexact nonlinear semi-infinite programming problems 51 For the third statement we need some slight generalizations of the Farkas– Minkowsky Lemma, in Girsanov (1972), for cones with non zero apex in a real normed space, having similar proof (see Bosch and Gómez 1997). Lemma 5 (Farkas–Minkowsky) : Let E1, E2 be normed spaces and K2 ⊂ E2, a convex cone with apex at x̄2 ∈ E2, such that θ2 ∈ cl(K2)\int(K2). Define on E1 the convex cone K1 = {x1 ∈ E1 : Ax1 ∈ K2} , where A : E1 −→ E2, is a continuous linear operator. Suppose there exists x̄1 ∈ E1 satisfying Ax̄1 ∈ int(K2). Then, the dual cone of K1 is computed by the formula: K ∗ 1 = A∗(K ∗ 2 ), where symbol A∗ denotes the adjoint operator of A. From this result we have: Lemma 6 If G(x̄, z̄) ∈ P\int(P) and RG = ∅ then: R∗ G = { f ∈ X∗ | f = (q∗ ◦ DG(x̄, z̄), θ∗ ∞), q∗ ∈ P∗, q∗ (G(x̄, z̄)) = 0 } . Proof It’s clear that RG = QG × L∞, where: QG = { (d, k) ∈ Rn × Cq | DG(x̄, z̄)(d, k) ∈ −G(x̄, z̄)+ int(P) } then, we only need to calculate Q∗G . Note that, by assumptions, QG is a nonempty open convex cone and it has exactly the form the Farkas–Minkowsky Lemma 5 refers to, taking E1 = Rn × Cq , E2 = Cq , K1 = QG , A = DG(x̄, z̄) and K2 = −G(x̄, z̄) + int(P). The apex of this last cone is −G(x̄, z̄) ∈ −P and we have θC ∈ cl(K2)\int(K2). Therefore, Lemma 5 can be applied and we obtain: K ∗ 1 = Q∗G = DG(x̄, z̄)∗(int (K2)∗), or Q∗G = { f = q∗ ◦ DG(x̄, z̄) : q∗ ∈ [int (−G(x̄, z̄)+ P)]∗ . But by definition: q∗ ∈ [int (−G(x̄, z̄)+ P)]∗ ⇔ q∗ ∈ C∗ q : q∗(−G(x̄, z̄)+ λρ) ≥ 0, ∀λ > 0, ∀ρ ∈ int(P), and this is equivalent to the inequalities: q∗(G(x̄, z̄)) ≤ 0, q∗(ρ) ≥ 0, ∀ρ ∈ cl(P). This means q∗ ∈ P∗,which we shall write q∗ θ∗ C and, because G(x̄, z̄) ∈ P, we also have the equality q∗(G(x̄, z̄)) = 0. Hence, the dual cone has the form: Q∗G = { f = q∗ ◦ DG(x̄, z̄) : q∗ ∈ C∗ q , q∗ θ∗ C , q∗(G(x̄, z̄)) = 0 } . 52 J. A. Gómez and P. Bosch For the tangent cone: K H := {(d, k, r) ∈ X | DH(x̄, ȳ)(d, r) = θ∞} since: DH(x̄, ȳ)(d, r)(ε) = ∇x h(x̄, ε)d − r(ε), a.e. ε ∈ [0, 1], DH(x̄, ȳ) is a continuous linear operator with full range: R(DH(x̄, ȳ)) = DH(x̄, ȳ)(Rn × L∞) = L∞ then R(DH(x̄, ȳ)) is closed and the orthogonal of the kernel of DH(x̄, ȳ) is the range of the adjoint DH(x̄, ȳ)∗ (see Luenberger 1992), i.e. K ∗ H = (N [DH(x̄, ȳ)])⊥ = R [ DH(x̄, ȳ)∗ ] . Therefore: K ∗ H = { f ∈ X∗ | f = (p∗ ◦ Dx H(x̄, ȳ), θ∗ C , p∗ ◦ Dy H(x̄, ȳ)), p∗ ∈ L∗∞ } Finally, we compute the dual of the cone of decreasing directions K ∗ 0 . We can write: K0 = ⋂ c∈C0 {(d, k, r) ∈ X | ∇x f (x̄, c)(d) < 0} .