Fritz John Type Duality in Nondifferentiable Continuous Programming with Equality and Inequality Constraints

A Fritz John type dual for a nondifferentiable continuous programming problem with equality and inequality constraints which represent many realistic situations is formulated using Fritz John type optimality conditions instead of Karush-Kuhn-Tucker type conditions and thus does not require a regularity condition. Various duality results under suitable generalized convexity assumptions are derived. A pair of Fritz John type dual continuous programming with natural boundary conditions rather than fixed end points is also presented. Finally, it is pointed that our duality results can be considered as dynamic generalizations of those of a nondifferentiable nonlinear programming problem in the presence of equality and inequality constraints recently treated in the literature

An extension of Yuan's Lemma and its applications in optimization

We prove an extension of Yuan's Lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of [A. Baccari and A. Trad. On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints. SIAM J. Opt., 15(2):394--408, 2005], where the classical necessary second-order optimality condition is proved under the assumption that the set of Lagrange multipliers is a bounded line segment. We prove the result under the more general assumption that the hessian of the Lagrangian evaluated at the vertices of the Lagrange multiplier set is a matrix set with at most rank 2. We apply the results to prove the classical second-order optimality condition to problems with quadratic constraints and without constant rank of the jacobian matrix

Necessary optimality criteria in mathematical programming in the presence of differentiability

AbstractWe consider the problem of minimizing a function over a region defined by an arbitrary set, equality constraints, and constraints of the inequality type defined via a convex cone. Under some moderate convexity assumptions on the arbitrary set we develop Optimality criteria of the minimum principle type which generalize the Fritz John Optimality conditions. As a consequence of this result necessary Optimality criteria of the saddle point type drop out. Here convexity requirements on the functions are relaxed to convexity at the point under investigation. We then present the weakest possible constraint qualification which insures positivity of the lagrangian multiplier corresponding to the objective function

On constraint qualifications with generalized convexity and optimality conditions

This paper deals with a multiobjective programming problem involving both equality constraints in infinite dimensional spaces. It is shown that some constraint qualifications together with a condition of interior points are sufficient conditions for the invexity of constraint maps with respect to the same scale map. Under a new constraint qualification which involves an invexity condition and a generalized Slater condition a Kuhn-Tucker necessary condition is established.Invexity, scale, constraint qualification, nearly S-convelike mapping.

KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization

For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both problems are revealed as we consider the KKT approach for the nonsmooth bilevel program. It turns out that the new inclusion (constraint) which appears as a consequence of the partial subdifferential of the lower-level Lagrangian (PSLLL) places the KKT reformulation of the nonsmooth bilevel program in a new class of mathematical program with both set-valued and complementarity constraints. While highlighting some new features of this problem, we attempt here to establish close links with the standard optimistic bilevel program. Moreover, we discuss possible natural extensions for C-, M-, and S-stationarity concepts. Most of the results rely on a coderivative estimate for the PSLLL that we also provide in this paper