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 Journal of Global Optimization   [SJR: 0.992]   [H-I: 60]   [4 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1573-2916 - ISSN (Online) 0925-5001    Published by Springer-Verlag  [2335 journals]
• Enclosure of all index-1 saddle points of general nonlinear functions
• Authors: Dimitrios Nerantzis; Claire S. Adjiman
Pages: 451 - 474
Abstract: Abstract Transition states (index-1 saddle points) play a crucial role in determining the rates of chemical transformations but their reliable identification remains challenging in many applications. Deterministic global optimization methods have previously been employed for the location of transition states (TSs) by initially finding all stationary points and then identifying the TSs among the set of solutions. We propose several regional tests, applicable to general nonlinear, twice continuously differentiable functions, to accelerate the convergence of such approaches by identifying areas that do not contain any TS or that may contain a unique TS. The tests are based on the application of the interval extension of theorems from linear algebra to an interval Hessian matrix. They can be used within the framework of global optimization methods with the potential of reducing the computational time for TS location. We present the theory behind the tests, discuss their algorithmic complexity and show via a few examples that significant gains in computational time can be achieved by using these tests.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0430-8
Issue No: Vol. 67, No. 3 (2017)

• An eigenvalue decomposition based branch-and-bound algorithm for nonconvex
• Authors: Cheng Lu; Zhibin Deng; Qingwei Jin
Pages: 475 - 493
Abstract: Abstract In this paper, we propose a branch-and-bound algorithm for finding a global optimal solution for a nonconvex quadratic program with convex quadratic constraints (NQPCQC). We first reformulate NQPCQC by adding some nonconvex quadratic constraints induced by eigenvectors of negative eigenvalues associated with the nonconvex quadratic objective function to Shor’s semidefinite relaxation. Under the assumption of having a bounded feasible domain, these nonconvex quadratic constraints can be further relaxed into linear ones to form a special semidefinite programming relaxation. Then an efficient branch-and-bound algorithm branching along the eigendirections of negative eigenvalues is designed. The theoretic convergence property and the worst-case complexity of the proposed algorithm are proved. Numerical experiments are conducted on several types of quadratic programs to show the efficiency of the proposed method.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0436-2
Issue No: Vol. 67, No. 3 (2017)

• Analysis of budget for interdiction on multicommodity network flows
• Authors: Pengfei Zhang; Neng Fan
Pages: 495 - 525
Abstract: Abstract In this paper, we concentrate on computing several critical budgets for interdiction of the multicommodity network flows, and studying the interdiction effects of the changes on budget. More specifically, we first propose general interdiction models of the multicommodity flow problem, with consideration of both node and arc removals and decrease of their capacities. Then, to perform the vulnerability analysis of networks, we define the function F(R) as the minimum amount of unsatisfied demands in the resulted network after worst-case interdiction with budget R. Specifically, we study the properties of function F(R), and find the critical budget values, such as $$R_a$$ , the largest value under which all demands can still be satisfied in the resulted network even under the worst-case interdiction, and $$R_b$$ , the least value under which the worst-case interdiction can make none of the demands be satisfied. We prove that the critical budget $$R_b$$ for completely destroying the network is not related to arc or node capacities, and supply or demand amounts, but it is related to the network topology, the sets of source and destination nodes, and interdiction costs on each node and arc. We also observe that the critical budget $$R_a$$ is related to all of these parameters of the network. Additionally, we present formulations to estimate both $$R_a$$ and $$R_b$$ . For the effects of budget increasing, we present the conditions under which there would be extra capabilities to interdict more arcs or nodes with increased budget, and also under which the increased budget has no effects for the interdictor. To verify these results and conclusions, numerical experiments on 12 networks with different numbers of commodities are performed.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0422-8
Issue No: Vol. 67, No. 3 (2017)

• Multiple depot ring star problem: a polyhedral study and an exact
algorithm
• Authors: Kaarthik Sundar; Sivakumar Rathinam
Pages: 527 - 551
Abstract: Abstract The multiple depot ring-star problem (MDRSP) is an important combinatorial optimization problem that arises in optical fiber network design and in applications that collect data using stationary sensing devices and autonomous vehicles. Given the locations of a set of customers and a set of depots, the goal is to (i) find a set of simple cycles such that each cycle (ring) passes through a subset of customers and exactly one depot, (ii) assign each non-visited customer to a visited customer or a depot, and (iii) minimize the sum of the routing costs, i.e., the cost of the cycles and the assignment costs. We present a mixed integer linear programming formulation for the MDRSP and propose valid inequalities to strengthen the linear programming relaxation. Furthermore, we present a polyhedral analysis and derive facet-inducing results for the MDRSP. All these results are then used to develop a branch-and-cut algorithm to obtain optimal solutions to the MDRSP. The performance of the branch-and-cut algorithm is evaluated through extensive computational experiments on several classes of test instances.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0431-7
Issue No: Vol. 67, No. 3 (2017)

• Branch and bound algorithm with applications to robust stability
• Authors: Laleh Ravanbod; Dominikus Noll; Pierre Apkarian
Pages: 553 - 579
Abstract: Abstract We discuss a branch and bound algorithm for global optimization of NP-hard problems related to robust stability. This includes computing the distance to instability of a system with uncertain parameters, computing the minimum stability degree of a system over a given set of uncertain parameters, and computing the worst case $$H_\infty$$ norm over a given parameter range. The success of our method hinges (1) on the use of an efficient local optimization technique to compute lower bounds fast and reliably, (2) a method with reduced conservatism to compute upper bounds, and (3) the way these elements are favorably combined in the algorithm.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0424-6
Issue No: Vol. 67, No. 3 (2017)

• A coverage-based Box-Algorithm to compute a representation for
optimization problems with three objective functions
• Authors: Tobias Kuhn; Stefan Ruzika
Pages: 581 - 600
Abstract: Abstract In this paper, a method for computing a substitute of the nondominated set, a so-called representation system, is proposed for tricriteria optimization problems. Our algorithm is easy to implement, it is flexibly applicable to a wide range of problems, and it takes several quality measures which have been proposed in the literature into account. The resulting representation system satisfies provably certain quality levels and, thus, it may be utilized to make proper and confident decisions.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0425-5
Issue No: Vol. 67, No. 3 (2017)

• Covers and approximations in multiobjective optimization
• Authors: Daniel Vanderpooten; Lakmali Weerasena; Margaret M. Wiecek
Pages: 601 - 619
Abstract: Abstract Due to the growing interest in approximation for multiobjective optimization problems (MOPs), a theoretical framework for defining and classifying sets representing or approximating solution sets for MOPs is developed. The concept of tolerance function is proposed as a tool for modeling representation quality. This notion leads to the extension of the traditional dominance relation to $$t\hbox {-}$$ dominance. Two types of sets representing the solution sets are defined: covers and approximations. Their properties are examined in a broader context of multiple solution sets, multiple cones, and multiple quality measures. Applications to complex MOPs are included.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0426-4
Issue No: Vol. 67, No. 3 (2017)

• A polynomially solvable case of the pooling problem
• Authors: Natashia Boland; Thomas Kalinowski; Fabian Rigterink
Pages: 621 - 630
Abstract: Abstract Answering a question of Haugland, we show that the pooling problem with one pool and a bounded number of inputs can be solved in polynomial time by solving a polynomial number of linear programs of polynomial size. We also give an overview of known complexity results and remaining open problems to further characterize the border between (strongly) NP-hard and polynomially solvable cases of the pooling problem.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0432-6
Issue No: Vol. 67, No. 3 (2017)

• Relaxations and discretizations for the pooling problem
• Authors: Akshay Gupte; Shabbir Ahmed; Santanu S. Dey; Myun Seok Cheon
Pages: 631 - 669
Abstract: Abstract The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0434-4
Issue No: Vol. 67, No. 3 (2017)

• Heuristic solution approaches for the maximum minsum dispersion problem
• Authors: Anna Martínez-Gavara; Vicente Campos; Manuel Laguna; Rafael Martí
Pages: 671 - 686
Abstract: Abstract The Maximum Minsum Dispersion Problem (Max-Minsum DP) is a strongly NP-Hard problem that belongs to the family of equitable dispersion problems. When dealing with dispersion, the operations research literature has focused on optimizing efficiency-based objectives while neglecting, for the most part, measures of equity. The most common efficiency-based functions are the sum of the inter-element distances or the minimum inter-element distance. Equitable dispersion problems, on the other hand, attempt to address the balance between efficiency and equity when selecting a subset of elements from a larger set. The objective of the Max-Minsum DP is to maximize the minimum aggregate dispersion among the chosen elements. We develop tabu search and GRASP solution procedures for this problem and compare them against the best in the literature. We also apply LocalSolver, a commercially available black-box optimizer, to compare our results. Our computational experiments show that we are able to establish new benchmarks in the solution of the Max-Minsum DP.
PubDate: 2017-03-01
DOI: 10.1007/s10898-016-0429-1
Issue No: Vol. 67, No. 3 (2017)

• Bounds tightening based on optimality conditions for nonconvex
box-constrained optimization
• Authors: Yash Puranik; Nikolaos V. Sahinidis
Pages: 59 - 77
Abstract: Abstract First-order optimality conditions have been extensively studied for the development of algorithms for identifying locally optimal solutions. In this work, we propose two novel methods that directly exploit these conditions to expedite the solution of box-constrained global optimization problems. These methods carry out domain reduction by application of bounds tightening methods on optimality conditions. This scheme is implicit and avoids explicit generation of optimality conditions through symbolic differentation, which can be memory and time intensive. The proposed bounds tightening methods are implemented in the global solver BARON. Computational results on a test library of 327 problems demonstrate the value of our proposed approach in reducing the computational time and number of nodes required to solve these problems to global optimality.
PubDate: 2017-01-01
DOI: 10.1007/s10898-016-0491-8
Issue No: Vol. 67, No. 1-2 (2017)

• Bounds tightening based on optimality conditions for nonconvex
box-constrained optimization
• Authors: Yash Puranik; Nikolaos V. Sahinidis
Pages: 59 - 77
Abstract: Abstract First-order optimality conditions have been extensively studied for the development of algorithms for identifying locally optimal solutions. In this work, we propose two novel methods that directly exploit these conditions to expedite the solution of box-constrained global optimization problems. These methods carry out domain reduction by application of bounds tightening methods on optimality conditions. This scheme is implicit and avoids explicit generation of optimality conditions through symbolic differentation, which can be memory and time intensive. The proposed bounds tightening methods are implemented in the global solver BARON. Computational results on a test library of 327 problems demonstrate the value of our proposed approach in reducing the computational time and number of nodes required to solve these problems to global optimality.
PubDate: 2017-01-01
DOI: 10.1007/s10898-016-0491-8
Issue No: Vol. 67, No. 1-2 (2017)

• Solving linear optimization over arithmetic constraint formula
• Authors: Li Chen; Yinrun Lyu; Chong Wang; Jingzheng Wu; Changyou Zhang; Nasro Min-Allah; Jamal Alhiyafi; Yongji Wang
Abstract: Abstract Since Balas extended the classical linear programming problem to the disjunctive programming (DP) problem where the constraints are combinations of both logic AND and OR, many researchers explored this optimization problem under various theoretical or application scenarios such as generalized disjunctive programming (GDP), optimization modulo theories (OMT), robot path planning, real-time systems, etc. However, the possibility of combining these differently-described but form-equivalent problems into a single expression remains overlooked. The contribution of this paper is two folded. First, we convert the linear DP/GDP model, linear-arithmetic OMT problem and related application problems into an equivalent form, referred to as the linear optimization over arithmetic constraint formula (LOACF). Second, a tree-search-based algorithm named RS-LPT is proposed to solve LOACF. RS-LPT exploits the techniques of interval analysis and nonparametric estimation for reducing the search tree and lowering the number of visited nodes. Also, RS-LPT alleviates bad construction of search tree by backtracking and pruning dynamically. We evaluate RS-LPT against two most common DP/GDP methods, three state-of-the-art OMT solvers and the disjunctive transformation based method on optimization benchmarks with different types and scales. Our results favor RS-LPT as compared to existing competing methods, especially for large scale cases.
PubDate: 2017-02-08
DOI: 10.1007/s10898-017-0499-8

• Approximation guarantees of algorithms for fractional optimization
problems arising in dispatching rules for INDS problems
• Authors: Hongtan Sun; Thomas C. Sharkey
Abstract: Abstract In this paper, we provide approximation guarantees of algorithms for the fractional optimization problems arising in the dispatching rules from recent literature for Integrated Network Design and Scheduling problems. These fractional optimization problem are proved to be NP-hard. The approximation guarantees are based both on the number of arcs in the network and on the number of machines in the scheduling environment. We further demonstrate, by example, the tightness of the factors for these approximation algorithms.
PubDate: 2017-02-07
DOI: 10.1007/s10898-017-0498-9

• Supply chain network capacity competition with outsourcing: a variational
equilibrium framework
• Authors: Anna Nagurney; Min Yu; Deniz Besik
Abstract: Abstract This paper develops a supply chain network game theory framework with multiple manufacturers/producers, with multiple manufacturing plants, who own distribution centers and distribute their products, which are distinguished by brands, to demand markets, while maximizing profits and competing noncooperatively. The manufacturers also may avail themselves of external distribution centers for storing their products and freight service provision. The manufacturers have capacities associated with their supply chain network links and the external distribution centers also have capacitated storage and distribution capacities for their links, which are shared among the manufacturers and competed for. We utilize a special case of the Generalized Nash Equilibrium problem, known as a variational equilibrium, in order to formulate and solve the problem. A case study on apple farmers in Massachusetts is provided with various scenarios, including a supply chain disruption, to illustrate the modeling and methodological framework as well as the potential benefits of outsourcing in this sector.
PubDate: 2017-01-31
DOI: 10.1007/s10898-017-0497-x

• GOSAC: global optimization with surrogate approximation of constraints
• Authors: Juliane Müller; Joshua D. Woodbury
Abstract: Abstract We introduce GOSAC, a global optimization algorithm for problems with computationally expensive black-box constraints and computationally cheap objective functions. The variables may be continuous, integer, or mixed-integer. GOSAC uses a two-phase optimization approach. The first phase aims at finding a feasible point by solving a multi-objective optimization problem in which the constraints are minimized simultaneously. The second phase aims at improving the feasible solution. In both phases, we use cubic radial basis function surrogate models to approximate the computationally expensive constraints. We iteratively select sample points by minimizing the computationally cheap objective function subject to the constraint function approximations. We assess GOSAC’s efficiency on computationally cheap test problems with integer, mixed-integer, and continuous variables and two environmental applications. We compare GOSAC to NOMAD and a genetic algorithm (GA). The results of the numerical experiments show that for a given budget of allowed expensive constraint evaluations, GOSAC finds better feasible solutions more efficiently than NOMAD and GA for most benchmark problems and both applications. GOSAC finds feasible solutions with a higher probability than NOMAD and GOSAC.
PubDate: 2017-01-28
DOI: 10.1007/s10898-017-0496-y

• Coradiant sets and $$\varepsilon$$ ε -efficiency in multiobjective
optimization
Abstract: Abstract This paper studies $$\varepsilon$$ -efficiency in multiobjective optimization by using the so-called coradiant sets. Motivated by the nonlinear separation property for cones, a similar separation property for coradiant sets is investigated. A new notion, called Bishop–Phelps coradiant set is introduced and some appropriate properties of this set are studied. This paper also introduces the notions of $$\varepsilon$$ -dual and augmented $$\varepsilon$$ -dual for Bishop and Phelps coradiant sets. Using these notions, some scalarization and characterization properties for $$\varepsilon$$ -efficient and proper $$\varepsilon$$ -efficient points are proposed.
PubDate: 2017-01-05
DOI: 10.1007/s10898-016-0495-4

• Coradiant sets and $$\varepsilon$$ ε -efficiency in multiobjective
optimization
Abstract: Abstract This paper studies $$\varepsilon$$ -efficiency in multiobjective optimization by using the so-called coradiant sets. Motivated by the nonlinear separation property for cones, a similar separation property for coradiant sets is investigated. A new notion, called Bishop–Phelps coradiant set is introduced and some appropriate properties of this set are studied. This paper also introduces the notions of $$\varepsilon$$ -dual and augmented $$\varepsilon$$ -dual for Bishop and Phelps coradiant sets. Using these notions, some scalarization and characterization properties for $$\varepsilon$$ -efficient and proper $$\varepsilon$$ -efficient points are proposed.
PubDate: 2017-01-05
DOI: 10.1007/s10898-016-0495-4

• An approximate bundle method for solving nonsmooth equilibrium problems
• Authors: Fan-Yun Meng; Li-Ping Pang; Jian Lv; Jin-He Wang
Abstract: Abstract We present an approximate bundle method for solving nonsmooth equilibrium problems. An inexact cutting-plane linearization of the objective function is established at each iteration, which is actually an approximation produced by an oracle that gives inaccurate values for the functions and subgradients. The errors in function and subgradient evaluations are bounded and they need not vanish in the limit. A descent criterion adapting the setting of inexact oracles is put forward to measure the current descent behavior. The sequence generated by the algorithm converges to the approximately critical points of the equilibrium problem under proper assumptions. As a special illustration, the proposed algorithm is utilized to solve generalized variational inequality problems. The numerical experiments show that the algorithm is effective in solving nonsmooth equilibrium problems.
PubDate: 2017-01-04
DOI: 10.1007/s10898-016-0490-9

• An approximate bundle method for solving nonsmooth equilibrium problems
• Authors: Fan-Yun Meng; Li-Ping Pang; Jian Lv; Jin-He Wang
Abstract: Abstract We present an approximate bundle method for solving nonsmooth equilibrium problems. An inexact cutting-plane linearization of the objective function is established at each iteration, which is actually an approximation produced by an oracle that gives inaccurate values for the functions and subgradients. The errors in function and subgradient evaluations are bounded and they need not vanish in the limit. A descent criterion adapting the setting of inexact oracles is put forward to measure the current descent behavior. The sequence generated by the algorithm converges to the approximately critical points of the equilibrium problem under proper assumptions. As a special illustration, the proposed algorithm is utilized to solve generalized variational inequality problems. The numerical experiments show that the algorithm is effective in solving nonsmooth equilibrium problems.
PubDate: 2017-01-04
DOI: 10.1007/s10898-016-0490-9

• Recent advances on the interval distance geometry problem
• Authors: Douglas S. Gonçalves; Antonio Mucherino; Carlile Lavor; Leo Liberti
Abstract: Abstract We discuss a discretization-based solution approach for a classic problem in global optimization, namely the distance geometry problem (DGP). We focus our attention on a particular class of the DGP which is concerned with the identification of the conformation of biological molecules. Among the many relevant ideas for the discretization of the DGP in the literature, we identify the most promising ones and address their inherent limitations to application to this class of problems. The result is an improved method for estimating 3D structures of small proteins based only on the knowledge of some distance restraints between pairs of atoms. We present computational results showcasing the usefulness of the new proposed approach. Proteins act on living cells according to their geometric and chemical properties: finding protein conformations can be very useful within the pharmaceutical industry in order to synthesize new drugs.
PubDate: 2017-01-03
DOI: 10.1007/s10898-016-0493-6

• Recent advances on the interval distance geometry problem
• Authors: Douglas S. Gonçalves; Antonio Mucherino; Carlile Lavor; Leo Liberti
Abstract: Abstract We discuss a discretization-based solution approach for a classic problem in global optimization, namely the distance geometry problem (DGP). We focus our attention on a particular class of the DGP which is concerned with the identification of the conformation of biological molecules. Among the many relevant ideas for the discretization of the DGP in the literature, we identify the most promising ones and address their inherent limitations to application to this class of problems. The result is an improved method for estimating 3D structures of small proteins based only on the knowledge of some distance restraints between pairs of atoms. We present computational results showcasing the usefulness of the new proposed approach. Proteins act on living cells according to their geometric and chemical properties: finding protein conformations can be very useful within the pharmaceutical industry in order to synthesize new drugs.
PubDate: 2017-01-03
DOI: 10.1007/s10898-016-0493-6

• Global optimization of non-convex piecewise linear regression splines
• Authors: Nadia Martinez; Hadis Anahideh; Jay M. Rosenberger; Diana Martinez; Victoria C. P. Chen; Bo Ping Wang
Abstract: Abstract Multivariate adaptive regression spline (MARS) is a statistical modeling method used to represent a complex system. More recently, a version of MARS was modified to be piecewise linear. This paper presents a mixed integer linear program, called MARSOPT, that optimizes a non-convex piecewise linear MARS model subject to constraints that include both linear regression models and piecewise linear MARS models. MARSOPT is customized for an automotive crash safety system design problem for a major US automaker and solved using branch and bound. The solutions from MARSOPT are compared with those from customized genetic algorithms.
PubDate: 2017-01-02
DOI: 10.1007/s10898-016-0494-5

• Global optimization of non-convex piecewise linear regression splines
• Authors: Nadia Martinez; Hadis Anahideh; Jay M. Rosenberger; Diana Martinez; Victoria C. P. Chen; Bo Ping Wang
Abstract: Abstract Multivariate adaptive regression spline (MARS) is a statistical modeling method used to represent a complex system. More recently, a version of MARS was modified to be piecewise linear. This paper presents a mixed integer linear program, called MARSOPT, that optimizes a non-convex piecewise linear MARS model subject to constraints that include both linear regression models and piecewise linear MARS models. MARSOPT is customized for an automotive crash safety system design problem for a major US automaker and solved using branch and bound. The solutions from MARSOPT are compared with those from customized genetic algorithms.
PubDate: 2017-01-02
DOI: 10.1007/s10898-016-0494-5

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