Algorithms for the Solution of Multiparametric Mixed-Integer Nonlinear Optimization Problems

Abstract

In this paper we present novel theoretical and algorithmic developments for the solution of mixed-integer optimization problems involving uncertainty, which can be posed as multiparametric mixed-integer optimization models, where uncertainty is described by a set of parameters bounded between lower and upper bounds. In particular, we address convex nonlinear formulations involving (i) 0−1 integer variables and (ii) uncertain parameters appearing linearly and separately and present on the right-hand side of the constraints. The developments reported in this work are based upon decomposition principles where the problem is decomposed into two iteratively converging subproblems: (i) a primal and (ii) a master subproblem, representing valid parametric upper and lower bounds on the final solution, respectively. The primal subproblem is formulated by fixing the integer variables which results in a multiparametric nonlinear programming (mp-NLP) problem, which is solved by outer-approximating the nonlinear functions at a number of points in the space of uncertain parameters to derive linear profiles. The aim of the master subproblem is then to propose another set of integer solutions which are better than the integer solutions that have already been analyzed in the primal subproblem. This can be achieved by (i) introducing cuts, (ii) employing outer-approximation (OA) principles, or (iii) using generalized benders decomposition (GBD) fundamentals. The algorithm terminates when the master problem cannot identify a better integer solution.

Explicit model predictive control of hybrid systems and multiparametric mixed integer polynomial programming

“…Generally, a multiparametric programming problem is of the following form 19,[38][39][40][41][42][43][44] JðhÞ 5 min x f ðx; hÞ subject to : hðx; hÞ 5 0 gðx; hÞ 0 (1)…”

“…Algorithms for explicit solutions of multiparametric MINLPs are presented in Dua and Pistikopoulos. 41 The problem of NMPC of hybrid systems, for the case that the nonlinearities are in polynomial form, is considered in the present work. In this article, hybrid systems will be considered in the following form x ðt 1 1Þ 5 f ðxðtÞ; uðtÞ; dðtÞÞ (18) z ðtÞ 5 n ðxðtÞ; uðtÞ; dðtÞÞ (19) subject to h ðxðtÞ; uðtÞ; dðtÞÞ 0 (20) where xðtÞ represents the state vector, zðtÞ the output vector, uðtÞ the control inputs, and dðtÞ the auxiliary variables that are used to model the hybrid nature of the system and might be either continuous or discrete.…”

Linear Model Predictive Control via Multiparametric Programming

“…The main characteristic of multiparametric programming is its ability to obtain [1,4,15,19,20,26,28] (i) the objective and optimization variable as functions of the varying parameters, and (ii) the regions in the space of the parameters where these functions are valid.…”

“…Hence, as the operating conditions vary, one does not have to reoptimize for the new set of conditions, since the optimal solution is already available as a function of the operating conditions. Mathematically, all the above applications can be generally posed as a multiparametric mixed-integer nonlinear programming (mp-MINLP) problem [1,14,15]. Furthermore, a number of variations of the problem (1.1) have been developed and examined in [2,14].…”

Stochastic LQ Control with Probabilistic Constraints

“…Multi-parametric cases were first studied by Pistikopoulos, Aecedevo and Duo [2] 2. LITERATURE REVIEW [3], [4], based on the post-optimality sensitivity analysis results of Fiacco [5] and Gal [6]. We will only summarize the results of multi-parametric programming that used to solve (2.1.1) with (2.1.4), which we called MPQP(x 0 ) from now on.…”