In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of...

In the first part of the tutorial, we introduced the problem of unconstrained optimization, provided necessary and sufficient conditions for optimality of a solution to this problem, and described the gradient descent method for finding a (locally) optimal solution to a given unconstrained optimization problem. We now describe another method for unconstrained optimization, namely Newton’s metho...

This chapter is mainly about William Karush and his role in the Karush-KuhnTucker theorem of nonlinear programming. It tells the story of fundamental optimization results that he obtained in his master’s thesis: results that he neither published nor advertised and that were later independently rediscovered and published by Harold W. Kuhn and Albert W. Tucker. The principal result – which concer...

ABSTRACT This paper provides a short introduction to the Lagrangian duality in convex optimization. At first the topic is motivated by outlining the importance of convex optimization. After that mathematical optimization classes such as convex, linear and non-convex optimization, are defined. Later the Lagrangian duality is introduced. Weak and strong duality are explained and optimality condit...

Previous work has studied the classical joint economic lot size model as an adverse selection problem with asymmetric cost information. Solving this problem is challenging due to the presence of countervailing incentives and two-dimensional information asymmetry, under which the classical single-crossing condition does not need to hold. In the present work we advance the existing knowledge abou...

Our domain G = (0,L) is an interval of length L. The boundary ∂G = {0,L} are the two endpoints. We consider here as an example the case (DD) of Dirichlet boundary conditions: Dirichlet conditions at x = 0 and x = L. For other boundary conditions (NN), (DN), (ND) one can proceed similarly. In one dimension the Laplace operator is just the second derivative with respect to x: ∆u(x, t) = uxx(x, t)...

In this paper, Karush-Kuhn-Tucker type robust necessary optimality conditions for a nonsmooth interval-valued optimization problem (UCIVOP) are formulated using the concept of LU-optimal solution and generalized Slater constraint qualification (GRSCQ). These shown to be sufficient under convexity. The Wolfe Mond-Weir dual problems over cones convexity assumptions, usual duality results establis...

In this paper, we study the nonconvex nonsmooth optimization problem (P) of minimizing a tangentially convex function with inequality constraints where constraint functions are convex. This is done by using cone tangential subdifferentials together new qualification. Indeed, present qualification to guarantee that Karush-Kuhn-Tucker conditions necessary and sufficient for optimality (P). Moreov...

Abstract In this paper, we introduce a new concept of sets and class functions called E -univex functions, respectively. For an -differentiable function, the -univexity is introduced by generalizing several concepts generalized convexity earlier defined into optimization theory. addition, some properties are investigated. Further, also introduced. Then, sufficiency so-called -Karush–Kuhn–Tucker...

The approximate Karush-Kuhn-Tucker (AKKT) condition is introduced being a necessary of the local weak efficient solution for optimistic bilevel optimization problems with multiple objectives in upper-level problems. We transform multi-objective problem into single-level by means value function transformation or KKT transformation. then prove that AKKT point to be without any constraint qualific...