Читать книгу Multi-parametric Optimization and Control - Efstratios N. Pistikopoulos - Страница 62

The two key statements that need to be proven are the convexity of and . Consider two generic parameter values and let , and and be the corresponding optimal objective function values and optimizers. Additionally, let and define and . Then, since , , and are feasible and satisfy the constraints and . As these constraints are affine, they can be linearly combined to obtain , and therefore is feasible for the optimization problem (2.2). Since a feasible solution exists at , an optimal solution exists at and thus is convex.

The optimal solution at will be less than or equal to the feasible solution, i.e. and thus:

(2.9a)

(2.9b)

i.e. , , . This proves the convexity of and . The piecewise affine nature of and is a direct result from the fact that the boundary between two regions belongs to both regions. Since the optimum is unique, the optimizer and thus the optimal objective function value must be continuous across the boundary.

In addition to the fundamental properties derived in Theorem 2.1, it is possible to infer more structural information about the connections between the critical regions: