Lian-Sheng Zhang

An exact lower order penalty function and its smoothing in nonlinear programming

In this article, we consider a lower order penalty function and its ε-smoothing for an inequality constrained nonlinear programming problem. It is shown that any strict local minimum satisfying the second-order sufficiency condition for the original problem is a strict local minimum of the lower order penalty function with any positive penalty parameter. By using an ε-smoothing approximation to the lower order penalty function, we get a modified smooth global exact penalty function under mild assumptions.

Convexification and Concavification for a General Class of Global Optimization Problems

A kind of general convexification and concavification methods is proposed for solving some classes of global optimization problems with certain monotone properties. It is shown that these minimization problems can be transformed into equivalent concave minimization problem or reverse convex programming problem or canonical D.C. programming problem by using the proposed convexification and concavification schemes. The existing algorithms then can be used to find the global solutions of the transformed problems.

Global optimality conditions for quadratic 0-1 optimization problems

In the present work, we intend to derive conditions characterizing globally optimal solutions of quadratic 0-1 programming problems. By specializing the problem of maximizing a convex quadratic function under linear constraints, we find explicit global optimality conditions for quadratic 0-1 programming problems, including necessary and sufficient conditions and some necessary conditions. We also present some global optimality conditions for the problem of minimization of half-products.

Peeling Off a Nonconvex Cover of an Actual Convex Problem: Hidden Convexity

Convexity is, without a doubt, one of the most desirable features in optimization. Many optimization problems that are nonconvex in their original settings may become convex after performing certain equivalent transformations. This paper studies the conditions for such hidden convexity. More specifically, some transformation-independent sufficient conditions have been derived for identifying hidden convexity. The derived sufficient conditions are readily verifiable for quadratic optimization problems. The global minimizer of a hidden convex programming problem can be identified using a local search algorithm.

MONOTONIZATION IN GLOBAL OPTIMIZATION

A general monotonization method is proposed for converting a constrained programming problem with non-monotone objective function and monotone constraint functions into a monotone programming problem. An equivalent monotone programming problem with only inequality constraints is obtained via this monotonization method. Then the existing convexification and concavefication methods can be used to convert the monotone programming problem into an equivalent better-structured optimization problem.

Identification of Hidden Convex Minimization Problems

If a nonconvex minimization problem can be converted into an equivalent convex minimization problem, the primal nonconvex minimization problem is called a hidden convex minimization problem. Sufficient conditions are developed in this paper to identify such hidden convex minimization problems. Hidden convex minimization problems possess the same desirable property as convex minimization problems: Any lo- cal minimum is also a global minimum. Identification of hidden convex minimization problem extends the reach of global optimization.