A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems

Abstract

In this paper, we outline the foundations of a general global optimisation strategy for the solution of multilevel hierarchical and general decentralised multilevel problems, based on our recent developments on multi-parametric programming and control theory. The core idea is to recast each optimisation subproblem, present in the hierarchy, as a multi-parametric programming problem, with parameters being the optimisation variables belonging to the remaining subproblems. This then transforms the multilevel problem into single-level linear/convex optimisation problems. For decentralised systems, where more than one optimisation problem is present at each level of the hierarchy, Nash equilibrium is considered. A three person dynamic optimisation problem is presented to illustrate the mathematical developments.

Model-based multi-parametric programming strategies towards the integration of design, control and operational optimization

“…Otherwise, the user needs to determine whether the structure of the approximate model of the I/O data set is problematic and repeat the procedure. In the last decade, multi-parametric programming has received considerable attention due to its applicability to a wide range of optimization and engineering problems such as optimal control ( [34,307]), the integration of design and control (Chapter 6), multi-objective optimization ( [256,297]) and bilevel optimization ( [104,105]). This in return has led to the development of novel algorithms for several classes of multi-parametric programming problems, such as multi-parametric mixed-integer programming ( [98,255]), multi-parametric dynamic programming ( [48,295]) and even inverse multi-parametric programming ( [16,155,250]).…”

“…Multi-parametric programming has been applied to bilevel and multilevel optimization ( [91,104,105,170] …”

Approximate solution algorithm for multi-parametric non-convex programming problems with polyhedral constraints

“…Note that the assumptions in Theorem 3 ensure that the inverse of the Jaccobian of Equation (5) exists [4,6,3]. In other words, when M 0 is not invertible any violation of assumptions in Theorem 3 is easily detected.…”

“…In [6] Dua et al, have proposed an algorithm to solve Equ. (5) in the entire range of the varying parameters for general convex problems.…”

“…Algorithms and applications of multiparametric programming approaches have been extensively studied in the literature [1,2,4,5,6,7] but all existing algorithms are limited to convex multi-parametric programming problems even if practical problems may include nonconvex formulation. In the last decade, Pistikopous et al, [3] proposed algorithmic strategy for parametric non-convex programming problems, but in practice the proposed algorithm is costly and limited to bilinear terms in addition to linear terms in the objective function as well as in constraint sets and a single perturbation parameter.…”