Global optimization conditions for certain nonconvex minimization problems (original) (raw)

Global optimality conditions in non-convex optimization

2012

In this talk we are going to present recent results regarding global optimality conditions for general non-convex optimization problems. First we are going to discuss complexity issues regarding the existence of points satisfying optimality conditions and the connection to complementarity problems. In addition, we are going to discuss surprising connections between optimality conditions and continuous formulations of discrete optimization problems.

On duality for nonconvex minimization problems within the framework of abstract convexity

arXiv (Cornell University), 2021

By applying the perturbation function approach, we propose the Lagrangian and the conjugate duals for minimization problems of the sum of two, generally nonconvex, functions. The main tools are the Φ-convexity theory and minimax theorems for Φ-convex functions. We provide conditions ensuring zero duality gap and introduce Φ-Karush-Kuhn-Tucker conditions that characterize solutions to primal and dual problems. We also discuss the relationship between the dual problems introduced in the present investigation and some conjugate-type duals existing in the literature.

Optimality and ε Lagrangian Duality for a Nonconvex Programming Problem with an Infinite Number of Constraints

Journal of Optimization Theory and Applications, 2009

In this paper, ε-optimality conditions are given for a nonconvex programming problem which has an infinite number of constraints. The objective function and the constraint functions are supposed to be locally Lipschitz on a Banach space. In a first part, we introduce the concept of regular ε-solution and propose a generalization of the Karush-Kuhn-Tucker conditions. These conditions are up to ε and are obtained by weakening the classical complementarity conditions. Furthermore, they are satisfied without assuming any constraint qualification. Then, we prove that these conditions are also sufficient for ε-optimality when the constraints are convex and the objective function is ε-semiconvex. In a second part, we define quasisaddlepoints associated with an ε-Lagrangian functional and we investigate their relationships with the generalized KKT conditions. In particular, we formulate a Wolfe-type dual problem which allows us to present ε-duality theorems and relationships between the KKT conditions and regular ε-solutions for the dual. Finally, we apply these results to two important infinite programming problems: the cone-constrained convex problem and the semidefinite programming problem.

Existence for minimization in Banach space with some applications

Existence of a minimum is shown for a coercive, kc differentiable function restricted to a finite intersection of half spaces under a structural assumption on the GLteaux derivative. This abstract existence result is illustrated by two examples from optimal control theory for distributed systems. 7'

On existence of solutions to non-convex minimization problems

arXiv (Cornell University), 2024

We provide new sufficient conditions for the finiteness of the optimal value and existence of solutions to a general problem of minimizing a proper closed function over a nonempty closed set. The conditions require an asymptotically bounded decay of a function, a relaxation of p-supercoercivity, and a certain relation for the asymptotic cone of the constraint set and the asymptotic function of the objective function. Our analysis combines these conditions with a regularization technique. We refine the notion of retractive directions of a set, extend its definition to functions, and establish some basic relations for such directions for both sets and functions. Using these tools, we provide existence of solutions results that generalize many of the results in the literature for both nonconvex and convex problems.