scholarly journals Generalized Benders Decomposition Method to Solve Big Mixed-Integer Nonlinear Optimization Problems with Convex Objective and Constraints Functions

Energies ◽  
2021 ◽  
Vol 14 (20) ◽  
pp. 6503
Author(s):  
Andrzej Karbowski
Keyword(s):  
Nonlinear Optimization ◽  
Benders Decomposition ◽  
Optimization Problems ◽  
Mixed Integer ◽  
Master Problem ◽  
Computational Algorithms ◽  
Generalized Benders Decomposition ◽  
Representation Theorems ◽  
Mixed Integer Nonlinear Optimization ◽  
Basic Approaches

The paper presents the Generalized Benders Decomposition (GBD) method, which is now one of the basic approaches to solve big mixed-integer nonlinear optimization problems. It concentrates on the basic formulation with convex objectives and constraints functions. Apart from the classical projection and representation theorems, a unified formulation of the master problem with nonlinear and linear cuts will be given. For the latter case the most effective and, at the same time, easy to implement computational algorithms will be pointed out.

Mixed-Integer Nonlinear Optimization

1995 ◽  
Author(s):  
Christodoulos A. Floudas
Keyword(s):  
Nonlinear Programming ◽  
Nonlinear Optimization ◽  
Benders Decomposition ◽  
Optimization Problems ◽  
Outer Approximation ◽  
Mixed Integer ◽  
Continuous Variables ◽  
Input Output ◽  
Mixed Integer Nonlinear Optimization ◽  
Continuous Domain

This chapter presents the fundamentals and algorithms for mixed-integer nonlinear optimization problems. Sections 6.1 and 6.2 outline the motivation, formulation, and algorithmic approaches. Section 6.3 discusses the Generalized Benders Decomposition and its variants. Sections 6.4, 6.5 and 6.6 presents the Outer Approximation and its variants with Equality Relaxation and Augmented Penalty. Section 6.7 discusses the Generalized Outer Approximation while section 6.8 compares the Generalized Benders Decomposition with the Outer Approximation. Finally, section 6.9 discusses the Generalized Cross Decomposition. A wide range of nonlinear optimization problems involve integer or discrete variables in addition to the continuous variables. These classes of optimization problems arise from a variety of applications and are denoted as Mixed-Integer Nonlinear Programming MINLP problems. The integer variables can be used to model, for instance, sequences of events, alternative candidates, existence or nonexistence of units (in their zero-one representation), while discrete variables can model, for instance, different equipment sizes. The continuous variables are used to model the input-output and interaction relationships among individual units/operations and different interconnected systems. The nonlinear nature of these mixed-integer optimization problems may arise from (i) nonlinear relations in the integer domain exclusively (e.g., products of binary variables in the quadratic assignment model), (ii) nonlinear relations in the continuous domain only (e.g., complex nonlinear input-output model in a distillation column or reactor unit), (iii) nonlinear relations in the joint integer-continuous domain (e.g., products of continuous and binary variables in the scheduling/ planning of batch processes, and retrofit of heat recovery systems). In this chapter, we will focus on nonlinearities due to relations (ii) and (iii). An excellent book that studies mixed-integer linear optimization, and nonlinear integer relationships in combinatorial optimization is the one by Nemhauser and Wolsey (1988). The coupling of the integer domain with the continuous domain along with their associated nonlinearities make the class of MINLP problems very challenging from the theoretical, algorithmic,and computational point of view. Apart from this challenge, however, there exists a broad spectrum of applications that can be modeled as mixed-integer nonlinear programming problems.

scholarly journals Solving mixed-integer nonlinear optimization problems using simultaneous convexification: a case study for gas networks

2021 ◽  
Author(s):  
Frauke Liers ◽  
Alexander Martin ◽  
Maximilian Merkert ◽  
Nick Mertens ◽  
Dennis Michaels
Keyword(s):  
Nonlinear Optimization ◽  
Optimization Problems ◽  
Mixed Integer ◽  
Global Optimality ◽  
Convex Relaxations ◽  
Convex Envelopes ◽  
Constraint Function ◽  
Convex Underestimators ◽  
Solution Approach ◽  
Mixed Integer Nonlinear Optimization

AbstractSolving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations of the feasible set. Known solution approaches based on spatial branch-and-bound become more effective the tighter the used relaxations are. Relaxations are commonly established by convex underestimators, where each constraint function is considered separately. Instead, a considerably tighter relaxation can be found via so-called simultaneous convexification, where convex underestimators are derived for more than one constraint function at a time. In this work, we present a global solution approach for solving mixed-integer nonlinear problems that uses simultaneous convexification. We introduce a separation method that relies on determining the convex envelope of linear combinations of the constraint functions and on solving a nonsmooth convex problem. In particular, we apply the method to quadratic absolute value functions and derive their convex envelopes. The practicality of the proposed solution approach is demonstrated on several test instances from gas network optimization, where the method outperforms standard approaches that use separate convex relaxations.

scholarly journals A mathematical justification for metronomic chemotherapy in oncology

2021 ◽  
Author(s):  
Luis Alberto Fernández Fernández ◽  
Cecilia Pola ◽  
Judith Sáinz-Pardo:
Keyword(s):  
Nonlinear Optimization ◽  
Pharmacokinetic Model ◽  
Optimization Problems ◽  
Metronomic Chemotherapy ◽  
Point Of View ◽  
Subject Classification ◽  
Mixed Integer ◽  
A Posteriori ◽  
Mixed Integer Nonlinear Optimization ◽  
Mathematical Justification

Abstract We mathematically justify metronomic chemotherapy as the best strategy to apply most cytotoxic drugs in oncology for both curative and palliative approaches, assuming the classical pharmacokinetic model together with the Emax pharmacodynamic and the Norton-Simon hypothesis.From the mathematical point of view, we will consider two mixed-integer nonlinear optimization problems, where the unknowns are the number of the doses and the quantity of each one, adjusting the administration times a posteriori.Mathematics Subject Classification: 93C15, 92C50, 90C30

scholarly journals The cost of not knowing enough: mixed-integer optimization with implicit Lipschitz nonlinearities

2021 ◽  
Author(s):  
Martin Schmidt ◽  
Mathias Sirvent ◽  
Winnifried Wollner
Keyword(s):  
Nonlinear Optimization ◽  
Optimization Problems ◽  
Nonlinear Function ◽  
Mixed Integer ◽  
Future Research ◽  
Integer Optimization ◽  
Worst Case ◽  
Mixed Integer Nonlinear Optimization ◽  
One Step ◽  
The Cost

AbstractIt is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function—in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute approximate global optimal solutions in finite time and we also provide a worst-case iteration bound. In some first numerical experiments we show that the “cost of not knowing enough” is rather high by comparing our approach with the open-source global solver . This reveals that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.

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