Optimization problems with quasiconvex inequality constraints

The constrained optimization problem min f(x), gj(x) 0 (j = 1, . . . , p) is considered, where f : X ! R and gj : X ! R are nonsmooth functions with domain X Rn. First-order necessary and first-order sufficient optimality conditions are obtained when gj are quasiconvex functions. Two are the main features of the paper: to treat nonsmooth problems it makes use of the Dini derivative; to obtain more sensitive conditions, it admits directionally dependent multipliers. The two cases, where the Lagrange function satisfies a non-strict and a strict inequality, are considered. In the case of a non-strict inequality pseudoconvex functions are involved and in their terms some properties of the convex programming problems are generalized. The efficiency of the obtained conditions is illustrated on an example. Key words: Nonsmooth optimization, Dini directional derivatives, quasiconvex functions, pseudoconvex functions, quasiconvex programming, Kuhn-Tucker conditions.

Nonsmooth analysis and optimization.

Huang Liren.Thesis (Ph.D.)--Chinese University of Hong Kong, 1993.Includes bibliographical references (leaves 96).Abstract --- p.1Introduction --- p.2References --- p.5Chapter Chapter 1. --- Some elementary results in nonsmooth analysis and optimization --- p.6Chapter 1. --- "Some properties for ""lim sup"" and ""lim inf""" --- p.6Chapter 2. --- The directional derivative of the sup-type function --- p.8Chapter 3. --- Some results in nonsmooth analysis and optimization --- p.12References --- p.19Chapter Chapter 2. --- On generalized second-order derivatives and Taylor expansions in nonsmooth optimization --- p.20Chapter 1. --- Introduction --- p.20Chapter 2. --- "Dini-directional derivatives, Clark's directional derivatives and generalized second-order directional derivatives" --- p.20Chapter 3. --- On Cominetti and Correa's conjecture --- p.28Chapter 4. --- Generalized second-order Taylor expansion --- p.36Chapter 5. --- Detailed proof of Theorem 2.4.2 --- p.40Chapter 6. --- Corollaries of Theorem 2.4.2 and Theorem 2.4.3 --- p.43Chapter 7. --- Some applications in optimization --- p.46Ref erences --- p.51Chapter Chapter 3. --- Second-order necessary and sufficient conditions in nonsmooth optimization --- p.53Chapter 1. --- Introduction --- p.53Chapter 2. --- Second-order necessary and sufficient conditions without constraint --- p.56Chapter 3. --- Second-order necessary conditions with constrains --- p.66Chapter 4. --- Sufficient conditions theorem with constraints --- p.77References --- p.87Appendix --- p.89References --- p.9

Higher-order conditions for strict efficiency revisited

D. V. Luu and P. T. Kien propose in Soochow J. Math. 33 (2007), 17-31, higher- order conditions for strict efficiency of vector optimization problems based on the derivatives introduced by I. Ginchev in Optimization 51 (2002), 47-72. These derivatives are defined for scalar functions and in their terms necessary and sufficient conditions can be obtained a point to be strictly efficient (isolated) minimizer of a given order for quite arbitrary scalar function. Passing to vector functions, Luu and Kien lose the peculiarity that the optimality conditions work with arbitrary functions. In the present paper, applying the mentioned derivatives for the scalarized problem and restoring the original idea, optimality conditions for strictly efficiency of a given order are proposed, which work with quite arbitrary vector functions. It is shown that the results of Luu and Kien are corollaries of the given conditions. Key words: nonsmooth vector optimization, higher-order optimality conditions, strict efficiency, isolated minimizers.

On local quasi efficient solutions for nonsmooth vector optimization

We are interested in local quasi efficient solutions for nonsmooth vector optimization problems under new generalized approximate invexity assumptions. We formulate necessary and sufficient optimality conditions based on Stampacchia and Minty types of vector variational inequalities involving Clarke's generalized Jacobians. We also establish the relationship between local quasi weak efficient solutions and vector critical points

First-Order Conditions for Optimization Problems with Quasiconvex Inequality Constraints

2000 Mathematics Subject Classification: 90C46, 90C26, 26B25, 49J52.The constrained optimization problem min f(x), gj(x) ≤ 0 (j = 1,…p) is considered, where f : X → R and gj : X → R are nonsmooth functions with domain X ⊂ Rn. First-order necessary and first-order sufficient optimality conditions are obtained when gj are quasiconvex functions. Two are the main features of the paper: to treat nonsmooth problems it makes use of Dini derivatives; to obtain more sensitive conditions, it admits directionally dependent multipliers. The two cases, where the Lagrange function satisfies a non-strict and a strict inequality, are considered. In the case of a non-strict inequality pseudoconvex functions are involved and in their terms some properties of the convex programming problems are generalized. The efficiency of the obtained conditions is illustrated on examples

Generalized Newton's Method based on Graphical Derivatives

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and $B$-differentiable versions of Newton's method for nonsmooth Lipschitzian equations