chapter  7
58 Pages

Vector Optimization: Basic Concepts and Solution Methods

Babes¸-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7.2 Mathematical Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

7.2.1 Partial Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 7.2.2 Increasing Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 7.2.3 Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 7.2.4 Biggest Weakly Monotone Functions . . . . . . . . . . . . . . . . . . . . 259

7.3 Pareto Maximality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7.3.1 Maximality with Respect to Extended Orders . . . . . . . . . . 262 7.3.2 Maximality of Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.3.3 Proper Maximality and Weak Maximality . . . . . . . . . . . . . . 263 7.3.4 Maximal Points of Free Disposal Hulls . . . . . . . . . . . . . . . . . . 266

7.4 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.4.1 The Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.4.2 Generalization to Order-Complete Sets . . . . . . . . . . . . . . . . . 269 7.4.3 Existence via Monotone Functions . . . . . . . . . . . . . . . . . . . . . . 271

7.5 Vector Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 7.5.1 Scalarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

7.6 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 7.6.1 Differentiable Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 7.6.2 Lipschitz Continuous Problems . . . . . . . . . . . . . . . . . . . . . . . . . 279 7.6.3 Concave Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.7 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 7.7.1 Weighting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 7.7.2 Constraint Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 7.7.3 Outer Approximation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Mathematical optimization studies the problem of finding a best element from a set of feasible alternatives with regard to a criterion or objective function. It is written in the form

optimize f(x)

subject to x ∈ X,

whereX is a nonempty set, called a feasible set or a set of feasible alternatives, and f is a real function on X , called a criterion or objective function. Here “optimize” stands for either “minimize” or “maximize,” which amounts to finding x¯ ∈ X such that either f(x¯) ≦ f(x) for all x ∈ X , or f(x¯) ≧ f(x) for all x ∈ X .