Second-order global optimality conditions for convex composite optimization

In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

[1]  Jean-Paul Penot Optimality conditions in mathematical programming and composite optimization , 1994, Math. Program..

[2]  Eng Ung Choo,et al.  Pseudolinearity and efficiency , 1984, Math. Program..

[3]  R. Tyrrell Rockafellar,et al.  Second-Order Optimality Conditions in Nonlinear Programming Obtained by Way of Epi-Derivatives , 1989, Math. Oper. Res..

[4]  Roberto Cominetti,et al.  A generalized second-order derivative in nonsmooth optimization , 1990 .

[5]  Philippe Michel,et al.  Second-order moderate derivatives , 1994 .

[6]  A. Ioffe,et al.  On some recent developments in the theory of second order optimality conditions , 1988 .

[7]  Xiaoqi Yang,et al.  Convex composite multi-objective nonsmooth programming , 1993, Math. Program..

[8]  Hidefumi Kawaski,et al.  An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems , 1988 .

[9]  A. Ioffe Necessary and Sufficient Conditions for a Local Minimum. 3: Second Order Conditions and Augmented Duality , 1979 .

[10]  G. Alistair Watson,et al.  First and second order conditions for a class of nondifferentiable optimization problems , 1980, Math. Program..

[11]  R. Rockafellar First- and second-order epi-differentiability in nonlinear programming , 1988 .

[12]  Richard J. Caron,et al.  Unboundedness of a convex quadratic function subject to concave and convex quadratic constraints , 1992 .

[13]  Clermont Dupuis,et al.  An Efficient Method for Computing Traffic Equilibria in Networks with Asymmetric Transportation Costs , 1984, Transp. Sci..

[14]  R. Tyrrell Rockafellar,et al.  Lagrange Multipliers and Optimality , 1993, SIAM Rev..

[15]  Xiaoqi Yang,et al.  Generalized second-order directional derivatives and optimization with C , 1992 .

[16]  Olvi L. Mangasarian,et al.  Second- and higher-order duality in nonlinear programming☆ , 1975 .

[17]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[18]  A. Ioffe Variational analysis of a composite function: A formula for the lower second order epi-derivative☆ , 1991 .

[19]  J. Hiriart-Urruty,et al.  Generalized Hessian matrix and second-order optimality conditions for problems withC1,1 data , 1984 .

[20]  Jean-Philippe Vial,et al.  Strong and Weak Convexity of Sets and Functions , 1983, Math. Oper. Res..

[21]  A. Ioffe Necessary and Sufficient Conditions for a Local Minimum. 2: Conditions of Levitin–Miljutin–Osmolovskii Type , 1979 .

[22]  Xiaoqi Yang,et al.  Convex composite minimization withC1,1 functions , 1995 .

[23]  Xiaoqi Yang Generalized second-order directional derivatives and optimality conditions , 1995, Bulletin of the Australian Mathematical Society.

[24]  Thomas R. Jefferson,et al.  Composite convex programs , 1991 .

[25]  Marc Teboulle,et al.  Hidden convexity in some nonconvex quadratically constrained quadratic programming , 1996, Math. Program..

[26]  F. Clarke Optimization And Nonsmooth Analysis , 1983 .

[27]  Xiaoqi Yang Vector variational inequality and its duality , 1993 .

[28]  James V. Burke,et al.  Second order necessary and sufficient conditions for convex composite NDO , 1987, Math. Program..

[29]  Xiaoqi Yang,et al.  On second-order directional derivatives , 1996 .

[30]  Vaithilingam Jeyakumar Composite Nonsmooth Programming with Gâteaux Differentiability , 1991, SIAM J. Optim..

[31]  J. V. Burke,et al.  Optimality conditions for non-finite valued convex composite functions , 1992, Math. Program..

[32]  R. Tobin Sensitivity analysis for variational inequalities , 1986 .

[33]  J. Burke An exact penalization viewpoint of constrained optimization , 1991 .

[34]  Ekkehard W. Sachs,et al.  Generalized quasiconvex mappings and vector optimization , 1986 .