Quadratic Maximization over the Reachable Values Set of a Convergent Discrete-time Affine System : The diagonalizable case

TL;DR

This paper presents an efficient method for solving quadratic maximization problems over the reachable set of a convergent discrete-time affine system with a diagonalizable matrix, by reducing infinite constraints to a finite set.

Contribution

It introduces a novel algorithm that extracts a minimal finite set of quadratic programs to solve the original infinite problem, leveraging linear algebra for efficiency.

Findings

The algorithm guarantees optimality of the solution.

It reduces the number of quadratic programs needed to solve.

Validated on academic and randomly generated examples.

Abstract

In this paper, we solve a maximization problem where the objective function is quadratic and convex or concave and the constraints set is the reachable value set of a convergent discrete-time affine system. Moreover, we assume that the matrix defining the system is diagonalizable. The difficulty of the problem lies in the infinite sequence to handle in the constraint set. Equivalently, the problem requires to solve an infinite number of quadratic programs. Therefore, the main idea is to extract a finite of them and to guarantee that the resolution of the extracted problems provides the optimal value and a maximizer for the initial problem. The number of quadratic programs to solve has to be the smallest possible. Actually, we construct a family of integers that over-approximate the exact number of quadratic programs to solve using basic ideas of linear algebra. This family of integers is used in the final algorithm. A new computation of an integer of the family within the algorithm ensures a reduction of the number of loop iterations. The method proposed in the paper is illustrated on small academic examples. Finally, the algorithm is experimented on randomly generated instances of the problem.