Phan Thien Thach

Optimization on Low Rank Nonconvex Structures

Preface. Part I: Foundations. 1. Scope of Global Optimization. 2. Quasi-Convexity. 3. D.C. Functions and D.C. Sets. 4. Duality. 5. Low-Rank Nonconvex Structures. 6. Global Search Methods and Basic D.C. Optimization Algorithms. Part II: Methods and Algorithms. 7. Parametric Approaches in Global Optimization. 8. Multiplicative Programming Problems. 9. Monotonic Problems. 10. Decomposition Methods by Prices. 11. Dynamic Programming Algorithms in Global Optimization. Part III: Selected Applications. 12. Low Rank Nonconvex Quadratic Programming. 13. Continuous Location. 14. Design Centering and Related Geometric Problems. 15. Multiobjective and Bilevel Programming. References. Index.

Monotonic Optimization: Branch and Cut Methods

Monotonic optimization is concerned with optimization problems dealing with multivariate monotonic functions and differences of monotonic functions. For the study of this class of problems a general framework (Tuy, 2000a) has been earlier developed where a key role was given to a separation property of solution sets of monotonic inequalities similar to the separation property of convex sets. In the present paper the separation cut is combined with other kinds of cuts, called reduction cuts, to further exploit the monotonic structure. Branch and cuts algorithms based on an exhaustive rectangular partition and a systematic use of cuts have proved to be much more efficient than the original polyblock and copolyblock outer approximation algorithms.

Dual approach to minimization on the set of Pareto-optimal solutions

LetX* be the set of Pareto-optimal solutions of a multicriteria programming problem. We are interested in finding a vectorxεX* which minimizes another criterion. SinceX* is a nonconvex set, our problem is that of minimization over a nonconvex set. By exploiting the fact that the number of criteria is often very small compared with the number of variables, we use a dual approach to obtain a practical algorithm. We report preliminary numerical results on problems with up to 100 variables and 5 criteria.

Diewert-Crouzeix conjugation for general quasiconvex duality and applications

A complicated factor in quasiconvex duality is the appearance of extra parameters. In order to avoid these extra parameters, one often has to restrict the class of quasiconvex functions. In this paper, by using the Diewert-Crouzeix conjugation, we present a duality without an extra parameter for general quasiconvex minimization problem. As an application, we prove a decentralization by prices for the Von Neumann equilibrium problem.

The design centering problem as a D.C. programming problem

The following problem is studied: Given a compact setS inRn and a Minkowski functionalp(x), find the largest positive numberr for which there existsx ∈ S such that the set of ally ∈ Rn satisfyingp(y−x) ≤ r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this “design centering problem” can be reformulated as the minimization of some d.c. function (difference of two convex functions) overRn. In the case where, moreover,p(x) = (xTAx)1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.

Mathematical programs with a two-dimensional reverse convex constraint

We consider the problem min {f(x): x ∈ G, T(x) ∉ int D}, where f is a lower semicontinuous function, G a compact, nonempty set in ℝn, D a closed convex set in ℝ2 with nonempty interior and T a continuous mapping from ℝn to ℝ2. The constraint T(x) ∉ int D is a reverse convex constraint, so the feasible domain may be disconnected even when f, T are affine and G is a polytope. We show that this problem can be reduced to a quasiconcave minimization problem over a compact convex set in ℝ2 and hence can be solved effectively provided f, T are convex and G is convex or discrete. In particular we discuss a reverse convex constraint of the form 〈c, x〉 · 〈d, x〉≤1. We also compare the approach in this paper with the parametric approach.

Optimization of Polynomial Fractional Functions

A new approach is proposed for optimizing a polynomial fractional function under polynomial constraints, or more generally, a synomial fractional function under synomial constraints. The approach is based on reformulating the problem as the optimization of an increasing function under monotonic constraints.

Linear Approximations in a Dynamic Programming Approach for the Uncapacitated Single-Source Minimum Concave Cost Network Flow Problem in Acyclic Networks

We consider minimum concave cost flow problems in acyclic, uncapacitated networks with a single source. For these problems a dynamic programming scheme is developed. It is shown that the concave cost functions on the arcs can be approximated by linear functions. Thus the considered problem can be solved by a series of linear programs. This approximation method, whose convergence is shown, works particularly well, if the nodes of the network have small degrees. Computational results on several classes of networks are reported.

D.c. sets, d.c. functions and nonlinear equations

Ad.c. set is a set which is the difference of two convex sets. We show that any set can be viewed as the image of a d.c. set under an appropriate linear mapping. Using this universality we can convert any problem of finding an element of a given compact set in ℝn into one of finding an element of a d.c. set. On the basis of this approach a method is developed for solving a system of nonlinear equations—inequations. Unlike Newton-type methods, our method does not require either convexity, differentiability assumptions or an initial approximate solution.

A decomposition method using a pricing mechanism for min concave cost flow problems with a hierarchical structure

In this paper we develop a decomposition method using a pricing mechanism which has been widely applied to linear and convex programs for a class of nonconvex optimization problems that are min concave cost flow problems under directed, uncapacitated networks with a hierarchical structure.