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An Algorithm for Separable Nonconvex Programming Problems
Abstract
In this paper we present an algorithm for solving mathematical programming problems of the form: Find x - (x<sub>1</sub>,..., x<sub>n</sub>) to minimize \sum \varphi <sub>i</sub>(x<sub>i</sub>) subject to x \in G and l < x < L. Each \varphi <sub>i</sub> is assumed to be lower semicontinuous, possibly nonconvex, and G is assumed to be closed. The algorithm is of the branch and bound type and solves a sequence of problems in each of which the objective function is convex. These problems correspond to successive partitions of the feasible set. Two different rules for refining the partitions are considered; these lead to convergence of the algorithm under different requirements on the problem functions. Examples are given, and computational considerations are discussed.
... In contrast to local optimization methods, deterministic global optimization methods, e.g., interval-based branch and bound (b&b) algorithms [1], guarantee to find the global solution for a predefined tolerance for optimality in finite time [2,3]. These methods are more expensive in terms of computational effort than their local counterparts. ...
... B Jens Deussen deussen@stce.rwth-aachen.de 1 Informatik 12: Software and Tools for Computational Engineering, RWTH Aachen University, Aachen, Germany A function f is called partially separable (also: decomposable or block-separable) if it is of the form ...
... A function is called (fully) separable [5,6] if it is of the form (1) with n [ j] = 1, j ∈ {1, . . . , p}. ...
We introduce a generalization of separability for global optimization, presented in the context of a simple branch and bound method. Our results apply to continuously differentiable objective functions implemented as computer programs. A significant search space reduction can be expected to yield an acceleration of any global optimization method. We show how to utilize interval derivatives calculated by adjoint algorithmic differentiation to examine the monotonicity of the objective with respect to so called structural separators and how to verify the latter automatically.
... Therefore, global deterministic optimization solvers like BARON (Tawarmalani & Sahinidis (2005)), ANTIGONE (Misener & Floudas (2014)) and MAiNGO (Bongartz et al. (2018)) are required to guarantee finding the best solution to non-convex optimization problems. Typically, these solvers use the spatial Branch-and-Bound algorithm (James & Richard (1969)) to which all of the variables and constraints are given to build the convex relaxation. This process is called the full-space formulation. ...
... Branch-and-bound algorithm (B&B) is one of the most important approaches in deterministic global optimization (James & Richard (1969)). The B&B can be summarized in 3 steps, node selection, branching variable selection and branching point selection. ...
Heat pump systems (HPSs) compete with low-cost heating technologies like gas-fired boilers, especially because of their role in decarbonizing the building sector. To design the most efficient HPS, mathematical optimization can be employed using simulation models of the HPS. Considering system dynamics inside the model results in many interdependencies between the heat pump components and the refrigerant selection and, thus, to large-scale models that are hard to optimize. Therefore, we propose a method that enhances the simultaneous optimization of design and operation of electrically driven HPSs. The developed method uses artificial neural networks (ANNs) to surrogate the calculations of the thermophysical properties of the used refrigerant. Thanks to the embedded ANNs, we are able to test five fluids to find the best refrigerant not only thermodynamically but also economically and environmentally. These fluids are Isobutane (R600a), Propane (R290), Fluoroethane (R161), Difluoroethane (R152a) and R410A. The HPS is optimized using the deterministic global optimization (DGO) solver, MAiNGO. As a result, we are able to decrease the capital investment costs by about 45% and the annual electric costs by 31% to 64% based on the refrigerant type. The savings in
total costs reach 56% for R161 and only 46% for R600a. An investigation of flammability and toxicity for the five proposed refrigerants is also conducted to evaluate their environmental impact.
... For instance, Carrillo (1977) ; Falk & Hoffman (1976) computed successive underestimations of the concave function to solve the problem optimally. Branch-and-bound based approaches are also common to solve these problems, where the feasible region is partitioned into smaller parts using branching ( Falk & Soland, 1969;Horst, 1976;Ryoo & Sahinidis, 1996;Tawarmalani & Sahinidis, 2004 ). Other ideas for handling concavities are based on extreme point ranking ( Murty, 1968;Taha, 1973 ) or generalized Benders decomposition ( Floudas et al., 1989;Li et al., 2011 ). ...
... For the single source uncapacitated version of minimum concave cost network flow problem, Gallo et al. (1980b) ; Guisewite & Pardalos (1991) implicitly enumerate the spanning tree of the network. Benson (1985) ; Falk & Soland (1969 Wu et al. (2021) proposed a deterministic annealing neural network-based method and two neural networks to solve the multiple sourcing version of the production-transportation problem. The authors tested the method for problems with small dimensions. ...
... В 20 столiттi був розроблений опуклий аналiз спочатку в геометрiї, а пiзнiше в серединi столiття його почали використовувати в оптимiзацiї. В цей же час з'являються першi роботи присвяченi пошуку глобального екстремуму [1,2]. Для опуклих задач були розробленi практично ефективнi алгоритми оптимiзацiї, якi дозволяють легко знайти точку глобального екстремуму (опуклi задачi є унiмодальними). ...
This article provides an analysis of the practical effectiveness of the method of exact quadratic regularization. Significant computational experiments have been performed to solve the complex multi-modal test and practical problems. The results of computational experiments are compared with the best results obtained by existing methods of global optimization. Comparative analysis shows a much greater practical efficiency of the method of exact quadratic regularization.
... Researchers have reported several algorithms and heuristics for solving non-convex problems, but most of them do not scale well while solving high-dimensional instances. The following methods are some of the exact algorithms that have been reported to solve non-convex problems: piece-wise linear underestimation (Dasci & Verter 2001), branch and bound (Falk & Soland 1969, Soland 1971, Shectman & Sahinidis 1998, spatial branch and bound (McCormick 1976, Lee & Grossmann 2001, Belotti et al. 2009), branch and reduce (Ryoo & Sahinidis 1995, 1996, cutting plane algorithm (Tuy 1964, Taha 1973, Tawarmalani & Sahinidis 2005, hybrid method (Marsten & Morin 1978), reformulation-linearization based technique (Zetina et al. 2021), etc. Most of these algorithms are suited to solve either concave minimization problems or very specific kinds of non-convex problems. ...
In this paper, we propose an exact general algorithm for solving non-convex optimization problems, where the non-convexity arises due to the presence of an inverse S-shaped function. The proposed method involves iteratively approximating the inverse S-shaped function through piece-wise linear inner and outer approximations. In particular, the concave part of the inverse S-shaped function is inner-approximated through an auxiliary linear program, resulting in a bilevel program, which is reduced to a single level using KKT conditions before solving it using the cutting plane technique. To test the computational efficiency of the algorithm, we solve a facility location problem involving economies and dis-economies of scale for each of the facilities. The computational experiments indicate that our proposed algorithm significantly outperforms the previously reported methods. We solve non-convex facility location problems with sizes up to 30 potential facilities and 150 customers. Our proposed algorithm converges to the global optimum within a maximum computational time of 3 hours for 95 percent of the datasets. For almost 60 percent of the test cases, the proposed algorithm outperforms the benchmark methods by an order of magnitude. The paper ends with managerial insights on facility network design involving economies and dis-economies of scale. One of the important insights points out that it may be optimal to increase the number of production facilities operating under dis-economies of scale with an overall decrease in transportation costs.
We propose a new upper bounding procedure for global minimization problems with continuous variables and possibly nonconvex inequality and equality constraints. Upper bounds are crucial for standard termination criteria of spatial branch-and-bound (SBB) algorithms to ensure that they can enclose globally minimal values sufficiently well. However, whereas for most lower bounding procedures from the literature, convergence on smaller boxes is established, this does not hold for several methods to compute upper bounds even though they often perform well in practice. In contrast, our emphasis is on the convergence. We present a new approach to verify the existence of feasible points on boxes, on which upper bounds can then be determined. To this end, we resort to existing convergent feasibility verification approaches for purely equality and box constrained problems. By considering carefully designed modifications of subproblems based on the approximation of active index sets, we enhance such methods to problems with additional inequality constraints. We prove that our new upper bounding procedure finds sufficiently good upper bounds so that termination of SBB algorithms is guaranteed after a finite number of iterations. Our theoretical findings are illustrated by computational results on a large number of standard test problems. These results show that compared with interval Newton methods from the literature, our proposed method is more successful in feasibility verification for both, a full SBB implementation (42 instead of 26 test problems) and exhaustive sequences of boxes around known feasible points (120 instead of 29 test problems).
History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous.
Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0162 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0162 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .
Non-convex optimization problems belong to a class of classical nonlinear optimization problems, which are often difficult to solve. An optimization problem becomes non-convex due to the presence of non-convex functions in the objective function or constraints. A function is a convex function if its Hessian matrix is positive and semi-definite for all values; otherwise, it is a non-convex function. A Hessian matrix is called positive semi-definite when the eigenvalues of the matrix are non-negative. A non-convex function can be either a concave function or a function that is neither a concave nor a convex function. A concave function is always negative semi-definite, indicating that the eigenvalues of the matrix are non-positive. This chapter starts with a short introduction to non-convex problems, followed by a discussion on different non-convex problems arising in supply chain and finance. Thereafter, the authors discuss different algorithms used for solving non-convex problems. Finally, the chapter conclude with the limitations of different algorithms.
Due to the rapid development of algorithms and techniques that are used to deal with mixed integer nonlinear programming (MINLP) problems, many global MINLP solvers were introduced. In this paper, computational experiments were done to compare between the performances of five of these solvers. Some of these solvers do not support trigonometric functions. Therefore, piecewise linear approximation (PLA) is applied to problems having these function so the solvers can deal with these problems. Additional computational tests were performed on to show how PLA can be useful, even to some powerful global solvers.
The essential features of the branch-and-bound approach to constrained optimization are described, and several specific applications are reviewed. These include integer linear programming Land-Doig and Balas methods, nonlinear programming minimization of nonconvex objective functions, the traveling-salesman problem Eastman and Little, et al. methods, and the quadratic assignment problem Gilmore and Lawler methods. Computational considerations, including trade-offs between length of computation and storage requirements, are discussed and a comparison with dynamic programming is made. Various applications outside the domain of mathematical programming are also mentioned.
A wide variety of branch and bound algorithms have recently been described in the literature. This paper provides a generalized description of such algorithms. An objective is to demonstrate the wide applicability of branch and bound to combinatorial problems in general. Two existing algorithms are used as illustrations and a discussion of computational efficiency is included.