Continuous Relaxation of MINLP Problems by Penalty Functions: A Practical Comparison (original) (raw)
Related papers
An Exact Penalty Approach for Mixed Integer Nonlinear Programming Problems
American Journal of Operations Research, 2011
We propose an exact penalty approach for solving mixed integer nonlinear programming (MINLP) problems by converting a general MINLP problem to a finite sequence of nonlinear programming (NLP) problems with only continuous variables. We express conditions of exactness for MINLP problems and show how the exact penalty approach can be extended to constrained problems.
Global optimization of mixed-integer nonlinear problems
AIChE Journal, 2000
Two novel deterministic global optimization algorithms for nonconvex mixed-integer problems (MINLPs) are proposed, using the advances of the BB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure Mixed-Integer BB algorithm (SMIN-BB addresses problems with nonconvexities in the continuous variables and linear and mixed-bilinear participation of the binary variables. The General Structure Mixed-Integer BB algorithm (GMIN-BB), is applicable to a very general class of problems for which the continuous relaxation is twice continuously di erentiable. Both algorithms are developed using the concepts of branch-and-bound, but they di er in their approach to each of the required steps. The SMIN-BB algorithm is based on the convex underestimation of the continuous functions while the GMIN-BB algorithm is centered around the convex relaxation of the entire problem. Both algorithms rely on optimization or interval based variable bound updates to enhance e ciency. A series of medium-size engineering applications demonstrates the performance of the algorithms. Finally, a comparison of the two algorithms on the same problems highlights the value of algorithms which can handle binary or integer variables without reformulation.
A Penalty Approach for Solving Nonsmooth and Nonconvex MINLP Problems
Springer Proceedings in Mathematics & Statistics, 2018
This paper presents a penalty approach for globally solving nonsmooth and nonconvex mixed-integer nonlinear programming (MINLP) problems. Both integrality constraints and general nonlinear constraints are handled separately by hyperbolic tangent penalty functions. Proximity from an iterate to a feasible promising solution is enforced by an oracle penalty term. The numerical experiments show that the proposed oracle-based penalty approach is effective in reaching the solutions of the MINLP problems and is competitive when compared with other strategies.
Review of nonlinear mixed-integer and disjunctive programming techniques
Optimization and Engineering, 2002
This paper has as a major objective to present a unified overview and derivation of mixed-integer nonlinear programming (MINLP) techniques, Branch and Bound, Outer-Approximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.
Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Methods
Chemie Ingenieur Technik, 2014
This work presents a review of the applications of mixed-integer nonlinear programming (MINLP) in process systems engineering (PSE). A review on the main deterministic MINLP solution methods is presented, including an overview of the main MINLP solvers. Generalized disjunctive programming (GDP) is an alternative higher-level representation of MINLP problems. This work reviews some methods for solving GDP models, and techniques for improving MINLP methods through GDP. The paper also provides a high-level review of the applications of MINLP in PSE, particularly in process synthesis, planning and scheduling, process control and molecular computing.
A trajectory-based method for mixed integer nonlinear programming problems
Journal of Global Optimization, 2017
A local trajectory-based method for solving mixed integer nonlinear programming problems is proposed. The method is based on the trajectory-based method for continuous optimization problems. The method has three phases, each of which performs continuous minimizations via the solution of systems of differential equations. A number of novel contributions, such as an adaptive step size strategy for numerical integration and a strategy for updating the penalty parameter, are introduced. We have shown that the optimal value obtained by the proposed method is at least as good as the minimizer predicted by a recent definition of a mixed integer local minimizer. Computational results are presented, showing the effectiveness of the method. Keywords Trajectory-based method • Mixed integer nonlinear programming • System of ordinary differential equations • Neighborhood • Local minimizer • Subproblem • Pattern search The second author thanks Professor Tapio Westerlund of Abo Akademi, Finland, for introducing him to the subject.
Derivative-Free Methods for Mixed-Integer Constrained Optimization Problems
Journal of Optimization Theory and Applications, 2014
Methods which do not use any derivative information are becoming popular among researchers, since they allow to solve many real-world engineering problems. Such problems are frequently characterized by the presence of discrete variables, which can further complicate the optimization process. In this paper, we propose derivative-free algorithms for solving continuously differentiable Mixed Integer Non-Linear Programming problems with general nonlinear constraints and explicit handling of bound constraints on the problem variables. We use an exterior penalty approach to handle the general nonlinear constraints and a local search approach to take into account the presence of discrete variables. We show that the proposed algorithms globally converge to points satisfying different necessary optimality conditions. We report a computational experience and a comparison with a well-known derivative-free optimization software package, i.e., NOMAD, on a set of test problems. Furthermore, we employ the proposed methods and NOMAD to solve a real problem concerning the optimal design of an industrial electric motor. This allows to show that the method converging to the better extended stationary points obtains the best solution also from an applicative point of view.
We present two linearization-based algorithms for mixed-integer nonlinear programs (MINLPs) having a convex continuous relaxation. The key feature of these algorithms is that, in contrast to most existing linearization-based algorithms for convex MINLPs, they do not require the continuous relaxation to be defined by convex nonlinear functions. For example, these algorithms can solve to global optimality MINLPs with constraints defined by quasiconvex functions. The first algorithm is a slightly modified version of the LP/NLP-based branch-and-bouund ( LP/NLP-BB ) algorithm of Quesada and Grossmann, and is closely related to an algorithm recently proposed by Bonami et al. (Math Program 119:331–352, 2009). The second algorithm is a hybrid between this algorithm and nonlinear programming based branch-and-bound. Computational experiments indicate that the modified LP/NLP-BB method has comparable performance to LP/NLP-BB on instances defined by convex functions. Thus, this algorithm has the potential to solve a wider class of MINLP instances without sacrificing performance.
An alternating optimization approach for mixed discrete non-linear programming
Engineering Optimization, 2009
This article contributes to the development of the field of Alternating Optimization (AO) and general Mixed Discrete Non-Linear Programming (MDNLP) by introducing a new decomposition algorithm (AO-MDNLP) based on the Augmented Lagrangian Multipliers method. In the proposed algorithm, an iterative solution strategy is proposed by transforming the constrained MDNLP problem into two unconstrained components or units; one solving for the discrete variables, and another for the continuous ones. Each unit focuses on minimizing a different set of variables while the other type is frozen. During optimizing each unit, the penalty parameters and multipliers are consecutively updated until the solution moves towards the feasible region. The two units take turns in evolving
MAiNGO – McCormick-based Algorithm for mixed-integer Nonlinear Global Optimization
2018
MAiNGO is a deterministic global optimization software for solving mixed-integer nonlinear programs (MINLP). It is applicable to a wide range of MINLPs and has been shown to have computational advantages for classes of problems that admit reduced-space formulations. Furthermore, it can also serve as a framework for simulation and local optimization. Main algorithmic features of MAiNGO are the operation in the original variable space through the use of McCormick relaxations (i.e., no introduction of auxiliary variables) through MC++ (Chachuat et al., IFAC-PapersOnline 48 (2015), 981), custom relaxations for various functions (including several functions relevant to process systems engineering), and significant flexibility in model formulation. In addition to a basic branch-and-bound with some state-of-the-art bound tightening techniques like duality-based bound tightening and optimization-based bound tightening, it implements specialized heuristics for tightening McCormick relaxation...