Complete characterizations of stable Farkas’ lemma and cone-convex programming duality

We establish necessary and sufficient conditions for a stable Farkas’ lemma. We then derive necessary and sufficient conditions for a stable duality of a cone-convex optimization problem, where strong duality holds for each linear perturbation of a given convex objective function. As an application, we obtain stable duality results for convex semi-definite programs and convex second-order cone programs.

[1]  Bevil Milton Glover,et al.  A generalized Farkas lemma with applications to quasidifferentiable programming , 1982, Z. Oper. Research.

[2]  B. Craven.  Control and optimization , 2019, Mathematical Modelling of the Human Cardiovascular System.

[3]  Stephen P. Boyd,et al.  Applications of second-order cone programming , 1998 .

[4]  Teodor Precupanu.  Closedness conditions for the optimality of a family of non-convex optimization problems , 1984 .

[5]  Radu Ioan Bot,et al.  Farkas-Type Results With Conjugate Functions , 2005, SIAM J. Optim..

[6]  Gert Wanka,et al.  An alternative formulation for a new closed cone constraint qualification , 2006 .

[7]  V. Jeyakumar.  The strong conical hull intersection property for convex programming , 2006, Math. Program..

[8]  M. A. López-Cerdá,et al.  Linear Semi-Infinite Optimization , 1998 .

[9]  Miguel A. Goberna,et al.  From linear to convex systems: consistency, Farkas' Lemma and applications , 2006 .

[10]  V. Barbu,et al.  Convexity and optimization in banach spaces , 1972 .

[11]  E. Andersen,et al.  Notes on Duality in Second Order and p -Order Cone Optimization , 2002 .

[12]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..

[13]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[14]  Vaithilingam Jeyakumar,et al.  Characterizations of solution sets of convex vector minimization problems , 2006, Eur. J. Oper. Res..

[15]  Vaithilingam Jeyakumar,et al.  Nonlinear Extensions of Farkas' Lemma with Applications to Global Optimization and Least Squares , 1995, Math. Oper. Res..

[16]  J. Gwinner,et al.  Results of farkas type , 1987 .

[17]  Vaithilingam Jeyakumar,et al.  Sequential Lagrangian Conditions for Convex Programs with Applications to Semidefinite Programming , 2005 .

[18]  J. Gwinner,et al.  Corrigendum and addendum to “results of farkas type” this journal 9, 471–520 (1987) , 1989 .

[19]  Vaithilingam Jeyakumar,et al.  Inequality systems and global optimization , 1996 .

[20]  Joachim Gwinner,et al.  On weak closedness, coerciveness, and inf-sup theorems , 1989 .

[21]  V. Jeyakumar,et al.  Farkas Lemma: Generalizations , 2009, Encyclopedia of Optimization.