Optimality condition and quasi-conjugate duality with zero gap in nonconvex optimization
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Optimality condition and quasi-conjugate duality with zero gap in nonconvex optimization Pham Ngoc Anh1
· Tran Van Thang2
Received: 1 March 2019 / Accepted: 12 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this article, we present an optimality condition in the form of a generalized KKT for nonconvex scalar-minimization problems. On the basis of the optimality condition, we present a quasi-conjugate duality for nonconvex scalar-minimization and vectorminimization problems. The duality is symmetric and has zero gap. Keywords Quasi-conjugate duality · Vector-minimization · Weakly efficient set · Quasi-subgradient
1 Introduction It is well-known that quasi-conjugate duality has been used widely and properly in optimization theory. An important feature of this duality theory is that, with a suitable definition of the quasi-conjugate of a function, we obtain strong duality and duality relationship. The duality relationship often reveals important and useful properties for the theoretical and computational studies of the original problem [3,4,7,9,11–13]. In [15–17], Thach presented quasi-conjugate duality for nonconvex problems with the quasi-conjugate of a function f : Rn → R as f ∗ ( p) = − inf{ f (x) : p, x ≥ 1}, ∀ p ∈ Rn .
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Pham Ngoc Anh [email protected] Tran Van Thang [email protected]
1
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Electric Power University, Hanoi, Vietnam
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P. N. Anh, T. Van Thang
More recently, in [19, Proposition 2.1], by defining the quasi-conjugate of a function f : Rn+ → R as f ∗ ( p) =
1 , ∀ p ∈ Rn+ , sup{ f (x) : p, x ≤ 1, x ≥ 0}
(2)
Thach and Thang developed a quasi-conjugate duality for problem that is a maximization of the polyhedral concave nondecreasing and homogeneous functions. In [21], Thang has showed some properties of f ∗ as follows: – If f is positively homogeneous, then so is f ∗ ; – The function f ∗ , quasi-conjugate of f , is quasiconcave and increasing on Rn+ ; – If f is a continuous quasi-concave and increasing function on Rn+ , then f is also the quasi-conjugate of f ∗ on Rn+ ; – If f is a continuous function on Rn+ , then f ∗ is upper semi-continuous on Rn+ and lower semi-continuous on Rn++ . The quasi-conjugate duality scheme is applied for studying a specific nonlinear optimization problem under resource allocation constraints [1,5,20]. In [18], Thach developed a quasi-conjugate duality for a multiple objective problem which appears in the application of minimizing cost under a given demand constraint by using the quasi-conjugate of a function f : Rn+ → R as f ∗ (x) = min{t : p ∈ t L ∗ }, ∀ p ∈ Rn+ , where L = {x ∈ Rn+ : f (x) ≤ 1} and L ∗ = { p ∈ Rn+ : p T x ≤ 1, ∀x ∈ L}. The objectives are nondecreasing coercive continuous convex homogeneous finitevalued function defined on Rn+ . Thach obtained symmetric duality with zero gap and a duality equation that helps to characterize the (weakly) efficient solutions of the primal problem and the dual.
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