Читать книгу Multi-parametric Optimization and Control - Efstratios N. Pistikopoulos - Страница 47

The aim is to represent a union of polytopes as a single set of linear inequality constraints in order to seamlessly include them within multi‐parametric programming problems. However, in order to address the possible non‐convexity within unions of polytopes, the introduction of suitable binary variables is required. First, consider that a point if and only if there exists at least one such that . Thus, one binary variable is defined such that

(1.35a)

(1.35b)

Let and denote the th row and element of and , respectively. Then, the statement holds if and only if , . Thus, one binary variable per row of , is defined such that

(1.36a)

(1.36b)

(1.36c)

Based on [10,11], Eqs. (1.36a)–(1.36c) are reformulated as

(1.37a)

(1.37b)

(1.37c)

(1.37d)

where , . Thus, the final formulation of the union as a set of linear inequality constraints featuring binary variables is given as

(1.38)