In this paper we present a survey of generalizations of the celebrated Farkas’s lemma, starting from systems of linear inequalities to a broad variety of non-linear systems. We focus on the generalizations which are targeted towards applications in continuous optimization. We also briefly...

We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$ f + g + k\circ H \ge h$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo>+</mo> <mi>k</mi> <mo>∘</mo> <mi>H</mi> <mo>≥</mo> <mi>h</mi> </mrow> </math> </EquationSource> </InlineEquation>. The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$ f + g +...</equationsource></inlineequation></equationsource></equationsource></inlineequation>

It is known that convex programming problems with separable inequality constraints do not have duality gaps. However, strong duality may fail for these programs because the dual programs may not attain their maximum. In this paper, we establish conditions characterizing strong duality for convex...

In this paper we present necessary conditions for global optimality for polynomial problems with box or bivalent constraints using separable polynomial relaxations. We achieve this by first deriving a numerically checkable characterization of global optimality for separable polynomial problems...

We develop a duality theory for minimax fractional programming problems in the face of data uncertainty both in the objective and constraints. Following the framework of robust optimization, we establish strong duality between the robust counterpart of an uncertain minimax convex–concave...

In this paper we examine multi-objective linear programming problems in the face of data uncertainty both in the objective function and the constraints. First, we derive a formula for the radius of robust feasibility guaranteeing constraint feasibility for all possible scenarios within a...