Partial Exactness for the Penalty Function of Biconvex Programming

Biconvex programming (or inequality constrained biconvex optimization) is an important model in solving many engineering optimization problems in areas like machine learning and signal and information processing. In this paper, the partial exactness of the partial optimum for the penalty function of biconvex programming is studied. The penalty function is partially exact if the partial Karush–Kuhn–Tucker (KKT) condition is true. The sufficient and necessary partially local stability condition used to determine whether the penalty function is partially exact for a partial optimum solution is also proven. Based on the penalty function, an algorithm is presented for finding a partial optimum solution to an inequality constrained biconvex optimization, and its convergence is proven under some conditions.

Smoothing Partially Exact Penalty Function of Biconvex Programming

In this paper, a smoothing partial exact penalty function of biconvex programming is studied. First, concepts of partial KKT point, partial optimum point, partial KKT condition, partial Slater constraint qualification and partial exactness are defined for biconvex programming. It is proved that the partial KKT point is equal to the partial optimum point under the condition of partial Slater constraint qualification and the penalty function of biconvex programming is partially exact if partial KKT condition holds. We prove the error bounds properties between smoothing penalty function and penalty function of biconvex programming when the partial KKT condition holds, as well as the error bounds between objective value of a partial optimum point of smoothing penalty function problem and its [Formula: see text]-feasible solution. So, a partial optimum point of the smoothing penalty function optimization problem is an approximately partial optimum point of biconvex programming. Second, based on the smoothing penalty function, two algorithms are presented for finding a partial optimum or approximate [Formula: see text]-feasible solution to an inequality constrained biconvex optimization and their convergence is proved under some conditions. Finally, numerical experiments show that a satisfactory approximate solution can be obtained by the proposed algorithm.

Optimality conditions for non-Lipschitz vector problems with inclusion constraints

<span class="fontstyle0"><span style="font-size: small;">We use the concept of approximation introduced by D.T Luc et al. as<br />generalized derivative for non-Lipschitz vector functions to consider vector problems<br />with non-Lipschitz data under inclusion constraints. Some calculus of approximations<br />are presented. A necessary optimality condition, type of KKT condition, for local<br />efficient solutions of the problems is established under an assumption on regularity.<br />Applications and numerical examples are also given.</span></span>

Improved enhanced Fritz John condition and constraints qualifications using convexificators

In this paper, using convexificators we derive enhanced Fritz John optimality condition for nonsmooth optimization problems having equality, inequality and abstract set constraint. This necessary optimality condition provides some more information about the extremal point in terms of converging sequences towards it. Then we employ this optimality condition to study enhanced KKT condition and to define associated ∂^*- pseudonormality and ∂^*-quasinormality concepts in terms of convexificators. Later, sufficiency for ∂^*-pseudonormality and some more results based on these concepts are investigated.