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Farkas-type results for nonconvex systems and applications to optimization

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Phd thesis: Farkas-type results for nonconvex systems and applications to optimization...New versions of the Farkas lemma for cone-convex systems and for sublinearconvex systems are established under Slater-type constraint qualication conditions and in the absence of the lower semi-continuity and the closedness of functions and constrained sets involved.

Phd thesis

## Farkas-type results for nonconvex systems and applications to optimization

By

TRAN HONG MO

New versions of the Farkas lemma for cone-convex systems and for sublinearconvex systems are established under Slater-type constraint qualication conditions and in the absence of the lower semi-continuity and the closedness of functions and constrained sets involved.

An extended version of Hahn-Banach-Lagrange theorem (that generalizes the one in [62]) Is established and it is shown to be equivalent to the extended Farkas lemma for cone-convex systems and for sublinear-convex systems just obtained.

Our results lead to extensions of other fundamental theorems such as the sandwich theorem, the Mazur-Orlicz theorem, and also the Hahn-Banach theorem itself for the case involving extended sublinear functions (the situation where the celebrated Hahn-Banach theorem failed).

The results obtained are then applied to get duality results and optimality conditions for a class of composite problems involving sublinear-convex mappings.

As illustrative examples, we consider the penalty problems associated to convex programming problems, a formula for the conjugate of the supremum of a family (possibly in nite, not lower semi-continuous) Of convex functions, and a special class of problems inspired in the Fenchel duality theorem. The duality results for the later class of problems give rise to some generalized versions of the Fenchel duality theorem which extends the one proposed recently by S. Simons in [62]. Moreover, the problems are considered in normed spaces and in this setting, a duality result leads to a separation theorem for convex sets in normed spaces.

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Contents

1 Notations and Preliminaries

2 Farkas-type results for systems involving composite functions

2.1 Dual qualication conditions and their relations

2.1.1 Dual qualication conditions in purely algebraic setting

2.1.2 Dual qualication conditions in convex setting

2.2 Characterizations of dual conditions{GeneralizedMoreau-Rockafellar re-sults

2.2.1 Dual conditions characterizing generalized Moreau-Rockafellarresults

2.2.2 Special cases

2.3 Nonconvex Farkas-type results

2.3.1 Nonconvex Farkas-type results

2.3.2 Special cases

2.4 Applications

2.4.1 Alternative-type theorems

2.4.2 Set containments

2.4.3 Fenchel-Rockafellar duality formula

3 New versions of Farkas lemma and Hahn-Banach theorem under Slater-type conditions

3.1 New versions of the Farkas lemma under Slater-type conditions

3.1.1 Farkas lemma for cone-convex systems

3.1.2 Farkas lemma for sublinear-convex systems

3.2 New versions of the Hahn-Banach theorem under Slater-type conditions

3.2.1 Extended Hahn-Banach-Lagrange theorem

3.2.2 Extension of the Hahn-Banach theorem, the sandwich theorem, and the Mazur-Orlicz theorem

3.2.3 The equivalence between extended versions of the Farkas lemmaand the Hahn-Banach-Lagrange theorem

3.3 Applications to optimization and convex analysis

3.3.1 Generalized optimization problems involving sublinear-convex map-pings

3.3.2 A Special case - Penalty problem in convex programming

3.3.3 Generalized Fenchel duality theorem and a separation theorem.

3.3.4 A conjugate formula for the supremum of a family of convexfunctions

4 From Farkas lemma to Hahn-Banach theorem

4.1 Characterizing extended Farkas lemmas for cone-convex systems

4.2 Charactering extended Farkas lemmas for sublinear-convex systems

4.3 Characterizing extended Hahn-Banach theorems

4.4 The equivalence between new versions of the Farkas lemma and HahnBanach-Lagrange theorem

5 Sequential Farkas lemmas and approximate Hahn-Banach theorems

5.1 Sequential Farkas lemma for cone-convex systems

5.2 Sequential Farkas lemma for sublinear-convex systems

5.3 Approximate Hahn-Banach theorems

5.4 Sublinear-convex optimization problems without any constraint quali-cation conditions

5.5 An application: Limiting conjugate formula for the supremum of a familyof convex functions

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