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CONJUGATE DUALITY for GENERALIZED CONVEX OPTIMIZATION PROBLEMS Anulekha Dhara and Aparna Mehra

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CONJUGATE DUALITY for GENERALIZED CONVEX OPTIMIZATION PROBLEMS Anulekha Dhara and Aparna Mehra JOURNAL OF INDUSTRIAL AND Website: http://AIMsciences.org MANAGEMENT OPTIMIZATION Volume 3, Number 3, August 2007 pp. 415–427 CONJUGATE DUALITY FOR GENERALIZED CONVEX OPTIMIZATION PROBLEMS Anulekha Dhara and Aparna Mehra Department of Mathematics Indian Institute of Technology Delhi Hauz Khas, New Delhi-110016, India (Communicated by M. Mathirajan) Abstract. Equivalence between a constrained scalar optimization problem and its three conjugate dual models is established for the class of general- ized C-subconvex functions. Applying these equivalent relations, optimality conditions in terms of conjugate functions are obtained for the constrained multiobjective optimization problem. 1. Introduction and preliminaries. Two basic approaches have emerged in lit- erature for studying nonlinear programming duality namely the Lagrangian ap- proach and the Conjugate Function approach. The latter approach is based on the perturbation theory which in turn helps to study stability of the optimization prob- lem with respect to variations in the data parameters. That is, the optimal solution of the problem can be considered as a function of the perturbed parameters and the effect of the changes in the parameters on the objective value of the problem can be investigated. In this context, conjugate duality helps to analyze the geometric and the economic aspects of the problem. It is because of this reason that since its early inception conjugate duality has been extensively studied by several authors for convex programming problems (see, [1, 6] and references therein). The class of convex functions possess rich geometrical properties that enable us to examine the conjugation theory. However, in non convex case, many of these properties fail to hold thus making it difficult to develop conjugate duality for such classes of functions. Recently, an attempt has been made by Bot¸et al.[3] to establish the equivalence between the three conjugate duals and the corresponding primal prob- lem under nearly convexity assumptions. It is important to observe that in their study convexity of the closure of epigraph set plays a crucial role in developing the results. There are many other important classes of generalized convex functions for which this kind of duality theory is yet to be described. Proceeding in this vein, our aim in the present work is to obtain strong duality between a scalar constrained optimization problem and its three conjugate duals, namely, the Lagrange dual, the Fenchel dual and the Fenchel-Lagrange dual under generalized C-subconvexity 2000 Mathematics Subject Classification. Primary: 90C26, 90C46; Secondary: 26B25. Key words and phrases. Nonlinear optimization problems, conjugate duality, generalized C- subconvex function, theorem of alternative. The first author is supported by Senior Research Fellowship from Council of Scientific and Industrial Research, India. 415 416 ANULEKHA DHARA AND APARNA MEHRA restrictions besides suitable constraint qualification. The results are subsequently extended to the multiobjective case. We first present a general framework necessary for understanding and developing the main results. We denote by ℜ¯ the extended real line ℜ ∪ {−∞, +∞} and F : ℜn → ℜ¯. Consider the unconstrained optimization problem: inf F (x). (Pu) x∈ℜn For p∗ ∈ ℜn, suppose H(x) = p∗T x − b is the linear function majorized by F , i.e., p∗T x − F (x) ≦ b, ∀ x ∈ℜn. The greatest lower bound of b satisfying the above inequality as the function of p∗ is termed as the conjugate function of F . Formally, the conjugate function F ∗ : ℜn → ℜ¯ of F is defined as F ∗(p∗) = sup (p∗T x − F (x)). x∈ℜn Geometrically, −F ∗(p∗) is the intercept with the F-axis of the highest linear function having a vector of coefficient p∗ and lying below the function F . In order to formulate the conjugate theory we need to define the perturbation function. For this, let ℜm be the space of the perturbation variables. Consider the perturbation function φ : ℜn ×ℜm → ℜ¯ satisfying the property that φ(x, 0) = n m F (x), ∀ x ∈ ℜ . For q ∈ ℜ , the perturbed problem associated with (Pu) is given by inf φ(x, q). x∈ℜn We now turn our attention to the following constrained optimization problem inf f(x) subject to g(x) ∈−C (Pc) x ∈ X where f : ℜn → ℜ¯, g : ℜn → ℜk, C ⊆ ℜk is a nonempty closed convex cone and k n X = dom(f) ∩ {∩i=1dom(gi)}⊆ℜ . The notation dom(f) stands for the effective n domain of f, i.e., dom(f)= {x ∈ℜ : f(x) < +∞}. Denote the feasible set of (Pc) by S = {x ∈ X : g(x) ∈−C}. Then (Pc) can be restated as follows inf f(x). x∈S Define a function F : ℜn → ℜ¯ as f(x) x ∈ S F (x)= +∞ otherwise Consequently, (Pc) reduces to the unconstrained problem (Pu). The associated perturbation function φ satisfies the following property f(x) x ∈ S φ(x, 0) = +∞ otherwise The conjugate of φ is defined as φ∗ : ℜn ×ℜm → ℜ¯, φ∗(x∗, q∗) = sup ((x∗, q∗)T (x, q) − φ(x, q)) x∈ℜn, q∈ℜm = sup (x∗T x + q∗T q − φ(x, q)). x∈ℜn, q∈ℜm CONJUGATE DUALITY 417 The perturbation function φ and its conjugate function φ∗ will be used to investigate the conjugate duality results for (Pc). 2. Lagrange and Fenchel dualities. We first describe a few notations that will be used in the sequel. The dual cone of C, denoted by C∗, is given by C∗ = {c∗ ∈ k ∗T k ℜ : c c ≧ 0, ∀ c ∈ C}. We shall consider the ordering ≦C in ℜ induced by C k as y ≦C x iff x − y ∈ C, ∀ x, y ∈ ℜ . Let us recall the three different kinds of conjugate dual models associated with (Pc). n k 2.1. The Lagrange Dual. The Lagrangian perturbed function φL : ℜ ×ℜ → ℜ¯ is defined as f(x), x ∈ X, g(x) ≦ q φ (x, q)= C L +∞, otherwise k and q ∈ℜ is the perturbation variable. The conjugate of the function φL is ∗ ∗ ∗ ∗T ∗T φL(x , q ) = sup (x x + q q − f(x)) k x∈X, q∈ℜ , g(x)≦C q = sup (x∗T x + q∗T (s + g(x)) − f(x)), q − g(x)= s ∈ C x∈X, s∈C sup(x∗T x + q∗T g(x) − f(x)), q∗ ∈−C∗ = x∈X ( +∞, otherwise For x ∈ S, q∗ ∈ C∗ we have q∗T g(x) ≦ 0. Consequently inf (f(x)+ q∗T g(x)) ≦ inf f(x), x∈X x∈S which implies sup inf (f(x)+ q∗T g(x)) ≦ inf f(x), (1) q∗∈C∗ x∈X x∈S i.e., ∗ ∗ sup (−φL(0, −q )) ≦ inf f(x). q∗∈C∗ x∈S The above inequality yields the following form of the Lagrangian dual ∗T sup inf (f(x)+ q g(x)). (DL) q∗∈C∗ x∈X It may be noted that in the construction of the Lagrangian dual the perturbation k parameter q ∈ℜ is associated with the constraints of (Pc). 2.2. The Fenchel Dual. Another dual of (Pc) is the Fenchel dual which is formu- lated by introducing the linear perturbation in the objective function variable while the feasible set remains unaltered, i.e., f(x + p), x ∈ S φ (x, p)= F +∞, otherwise n where p ∈ℜ is the perturbation variable. The conjugate of the function φF is ∗ ∗ ∗ ∗T ∗T φF (x ,p ) = sup (x x + p p − f(x + p)) x∈S, p∈ℜn = sup (p∗T r − f(r)) + sup((x∗ − p∗)T x), x + p = r r∈ℜn x∈S =f ∗(p∗) − inf ((p∗ − x∗)T x). x∈S 418 ANULEKHA DHARA AND APARNA MEHRA Using the definition of conjugate function we have −f ∗(p∗)+ p∗T x ≦ f(x), ∀ p∗ ∈ℜn, ∀ x ∈ℜn. Thus sup (−f ∗(p∗) + inf p∗T x) ≦ inf f(x). (2) p∗∈ℜn x∈S x∈S This leads to the Fenchel dual problem of the following form ∗ ∗ ∗T sup (−f (p ) + inf p x). (DF ) p∗∈ℜn x∈S ∗T It is important to observe that in (DF ) the infimum of the linear function p x is taken over the feasible set S of (Pc) comprising of the constraint functions g. 2.3. The Fenchel-Lagrange Dual. Combining the ideas of the Lagrange dual and the Fenchel dual, Wanka and Bot¸[7] proposed the Fenchel-Lagrange dual in which the perturbation parameters appear both in the constraints as well as in the objective function variables. The Fenchel Lagrangian perturbed function φFL : ℜn ×ℜn ×ℜk → ℜ¯ is defined as f(x + p), x ∈ X, g(x) ≦ q φ (x, p, q)= C FL +∞, otherwise where (p, q) ∈ ℜn ×ℜk is the perturbation vector. The conjugate of the function φFL is ∗ ∗ ∗ ∗ ∗T ∗T ∗T φFL(x ,p , q ) = sup (x x + p p + q q − f(x + p)) n k x∈X, p∈ℜ , q∈ℜ , g(x)≦C q = sup (x∗T x + p∗T (r − x)+ q∗T (s + g(x)) − f(r)), x∈X, r∈ℜn, s∈C q − g(x)= s, x + p = r f ∗(p∗) + sup((x∗ − p∗)T x + q∗T g(x)), q∗ ∈−C∗ = x∈X ( +∞, otherwise The Fenchel-Lagrange dual problem associated with (Pc) is given by ∗ ∗ ∗T ∗T sup (−f (p ) + inf (p x + q g(x))). (DFL) p∗∈ℜn, q∗∈C∗ x∈X The feasibility conditions of (Pc) along with the definition of conjugate function yields f(x) ≧ −f ∗(p∗)+ p∗T x + q∗T g(x), ∀ x ∈ S, ∀ p∗ ∈ℜn, ∀ q∗ ∈ C∗ which implies inf f(x) ≧ −f ∗(p∗) + inf (p∗T x + q∗T g(x)) x∈S x∈S (3) ≧ −f ∗(p∗) + inf (p∗T x + q∗T g(x)), ∀ p∗ ∈ℜn, ∀ q∗ ∈ C∗. x∈X Thus leading to the Fenchel-Lagrange weak duality.
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