Farkas-Type Results With Conjugate Functions

We present some new Farkas-type results for inequality systems involving a finite as well as an infinite number of convex constraints. For this, we use two kinds of conjugate dual problems, namely an extended Fenchel-type dual problem and the recently introduced Fenchel--Lagrange dual problem. For the latter, which is a "combination" of the classical Fenchel and Lagrange duals, the strong duality is established.

[1]  Vaithilingam Jeyakumar,et al.  A general approach to dual characterizations of solvability of inequality systems with applications. , 1995 .

[2]  Strong Duality for Generalized Convex Optimization Problems , 2005 .

[3]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[4]  J. Farkas.  Theorie der einfachen Ungleichungen. , 1902 .

[5]  Gert Wanka,et al.  Farkas-type results for inequality systems with composed convex functions via conjugate duality , 2006 .

[6]  Olvi L. Mangasarian,et al.  Set Containment Characterization , 2002, J. Glob. Optim..

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

[8]  Vaithilingam Jeyakumar,et al.  Inequality systems and optimization , 1991 .

[9]  R. Reinhardt,et al.  Einführung in die nichtlineare optimierung , 1977 .

[10]  Vaithilingam Jeyakumar,et al.  Complete Characterizations of Global Optimality for Problems Involving the Pointwise Minimum of Sublinear Functions , 1996, SIAM J. Optim..

[11]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[12]  Gert Wanka,et al.  Multiobjective duality for convex ratios , 2002 .

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

[14]  Vaithilingam Jeyakumar,et al.  Characterizing global optimality for DC optimization problems under convex inequality constraints , 1996, J. Glob. Optim..

[15]  Sorin-Mihai Grad,et al.  Fenchel-Lagrange Duality Versus Geometric Duality in Convex Optimization , 2006 .

[16]  Constantin Zălinescu,et al.  A generalization of the Farkas lemma and applications to convex programming , 1978 .

[17]  Gert Wanka,et al.  Farkas-type Results for Max-functions and Applications , 2006 .

[18]  Gert Wanka,et al.  On the Relations Between Different Dual Problems in Convex Mathematical Programming , 2002 .

[19]  R. Boţ,et al.  Fenchel-Lagrange versus Geometric Duality in Convex Optimization , 2016 .

[20]  V. Jeyakumar,et al.  Characterizing Set Containments Involving Infinite Convex Constraints and Reverse-Convex Constraints , 2002, SIAM J. Optim..

[21]  Vaithilingam Jeyakumar,et al.  A Farkas lemma for difference sublinear systems and quasidifferentiable programming , 1994, Math. Program..

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

[23]  Chung-Wei Ha.  On systems of convex inequalities , 1979 .

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