Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

The augmented Lagrange multiplier as an important concept in duality theory for optimization problems is extended in this paper to generalized augmented Lagrange multipliers by allowing a nonlinear support for the augmented perturbation function. The existence of generalized augmented Lagrange multipliers is established by perturbation analysis. Meanwhile, the relations among generalized augmented Lagrange multipliers, saddle points, and zero duality gap property are developed.

Modified Courant-Beltrami penalty function and a duality gap for invex optimization problem

In this paper, we modified a Courant-Beltrami penalty function method for constrained optimization problem to study a duality for convex nonlinear mathematical programming problems. Karush-Kuhn-Tucker (KKT) optimality conditions for the penalized problem has been used to derived KKT multiplier based on the imposed additional hypotheses on the constraint function g. A zero-duality gap between an optimization problem constituted by invex functions with respect to the same function η and their Lagrangian dual problems has also been established. The examples have been provided to illustrate and proved the result for the broader class of convex functions, termed invex functions.

On Sufficient Global Optimality Conditions for Bivalent Quadratic Programs with Quadratic Constraints

We show the recent sufficient global optimality condition for the quadratic constrained bivalent quadratic optimization problem is equivalent to verify the zero duality gap. Then, based on the optimal parametric Lagrangian dual model, we establish improved sufficient conditions by strengthening the dual bound.