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On solving generalized convex MINLP problems using supporting hyperplane techniques

JOURNAL OF GLOBAL OPTIMIZATION 2018年第4期71卷 987-1011页

作者： Westerlund, Tapio Eronen, Ville-Pekka Makela, Marko M. Abo Akad Univ Fac Sci & Engn Turku Finland Univ Turku Dept Math & Stat Turku Finland

Solution methods for convex mixed integer nonlinear programming (MINLP) problems have, usually, proven convergence properties if the functions involved are differentiable and convex. For other classes of convex MINLP problems fewer results have been given. Classical differential calculus can, though, be generalized to more general classes of functions than differentiable, via subdifferentials and subgradients. In addition, more general than convex functions can be included in a convex problem if the functions involved are defined from convex level sets, instead of being defined as convex functions only. The notion generalized convex, used in the heading of this paper, refers to such additional properties. The generalization for the differentiability is made by using subgradients of Clarke's subdifferential. Thus, all the functions in the problem are assumed to be locally Lipschitz continuous. The generalization of the functions is done by considering quasiconvex functions. Thus, instead of differentiable convex functions, nondifferentiable -quasiconvex functions can be included in the actual problem formulation and a supporting hyperplane approach is given for the solution of the considered MINLP problem. Convergence to a global minimum is proved for the algorithm, when minimizing an -pseudoconvex function, subject to -pseudoconvex constraints. With some additional conditions, the proof is also valid for -quasiconvex functions, which sums up the properties of the method, treated in the paper. The main contribution in this paper is the generalization of the Extended Supporting Hyperplane method in Eronen et al. (J Glob Optim 69(2):443-459, 2017) to also solve problems with -pseudoconvex objective function.

关键词： Nonsmooth optimization mixed-integer nonlinear programming Generalized convexities Supporting hyperplanes Cutting planes

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Comprehensive approach for distribution system planning with uncertainties

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IET GENERATION TRANSMISSION & DISTRIBUTION 2019年第24期13卷 5467-5477页

作者： Seta, Felipe da Silva de Oliveira, Leonardo W. de Oliveira, Edimar J. Fed Ctr Technol Educ Dept Elect Engn Angra Dos Reis RJ Brazil Fed Univ Juiz Fora UFJF Dept Elect Engn Juiz De Fora MG Brazil

This study presents a comprehensive approach for the distribution system expansion planning (DSEP) that considers investment, operation, carbon dioxide emission and reliability costs, as well as uncertainties over load demand and wind-based distributed generation. A restoration strategy is taken into account for obtaining the energy not supplied under the 'N - 1' criterion and the corresponding reliability cost over a planning horizon. The problem is modelled as mixed-integer non-linear programming by using the artificial immune system algorithm. Also, two methods to represent uncertainties are applied and compared: an interval technique through an interval power flow and a scenario-based approach. Network constraints are considered, as the limits of current, voltage and power from substations, as well as the obtaining of radial and connected topology. The novelty of the proposed interval DSEP consists of handling the uncertainties over operation in an efficient manner through a single step, instead of the several deterministic evaluations of the scenario-based approach. Numerical results are presented for well known test systems, which show the potentials of the proposed approach.

关键词： power distribution planning distributed power generation investment load flow artificial immune systems power distribution economics nonlinear programming power distribution faults integer programming power distribution reliability mixed-integer nonlinear programming artificial immune system algorithm interval technique interval power flow scenario-based approach interval DSEP known test systems distribution system expansion planning carbon dioxide emission reliability costs uncertainties overload demand wind-based distributed generation restoration strategy planning horizon

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Reactive power planning using convex line-wise power balance equations for radial distribution systems

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IET GENERATION TRANSMISSION & DISTRIBUTION 2020年第12期14卷 2399-2406页

作者： Aldik, Abdelrahman Venkatesh, Bala Ryerson Univ Elect & Comp Engn Dept 350 Victoria St Toronto ON M5B 2K3 Canada

Optimal capacitor placement for radial distribution systems (RDSs) considers minimising the total cost of new fixed capacitors, switchable capacitors, and losses, while satisfying power balance equations, limits on bus voltages and capacitor limits. It is a non-convex mixed-integer non-linear programming (MINLP) challenge. In this study, the authors propose a solution method using a line-wise model (LWM) of power balance equations. First, equations for LWM are presented with their Jacobian for solving the power flow problem using Newton-Raphson method. Then, an optimal non-convex MINLP capacitor placement formulation with LWM power balance equations is presented. Thereafter, it is transformed into a convex mixed-integer conic programming formulation using second-order conic relaxation. Both the non-convex and convex optimal capacitor placement formulations are used to study 69-bus and 136-bus RDS. The results are compared with a formulation that uses the branch flow model (BFM) for power balance equations. Results show that the non-convex LWM-based formulation is twice as fast when compared with the BFM-based formulation. The convex LWM-based formulation is from 4 to 30 times as fast when compared with the BFM-based formulation, demonstrating the benefits of the use of the LWM-based formulation for enhancing the solution space of the optimisation problem.

关键词： nonlinear programming Newton-Raphson method distribution networks power capacitors linear programming convex programming load flow integer programming reactive power optimisation reactive power planning convex line-wise power balance equations radial distribution systems fixed capacitors switchable capacitors satisfying power balance equations capacitor limits mixed-integer nonlinear programming line-wise model power flow problem optimal capacitor placement formulation LWM power balance equations nonconvex MINLP challenge convex mixed-integer conic programming formulation convex optimal capacitor placement formulations nonconvex LWM-based formulation BFM-based formulation convex LWM-based formulation

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Perspective reformulations of mixed integer nonlinear programs with indicator variables

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MATHEMATICAL programming 2010年第1-2期124卷 183-205页

作者： Guenluek, Oktay Linderoth, Jeff IBM Corp Thomas J Watson Res Ctr Dept Math Sci Yorktown Hts NY 10598 USA Univ Wisconsin Dept Ind & Syst Engn Madison WI 53706 USA

We study mixed integer nonlinear programs (MINLP)s that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables. An indicator variable, when it is "turned off", forces some of the decision variables to assume fixed values, and, when it is "turned on", forces them to belong to a convex set. Many practical MINLPs contain integer variables of this type. We first study a mixed integer set defined by a single separable quadratic constraint and a collection of variable upper and lower bound constraints, and a convex hull description of this set is derived. We then extend this result to produce an explicit characterization of the convex hull of the union of a point and a bounded convex set defined by analytic functions. Further, we show that for many classes of problems, the convex hull can be expressed via conic quadratic constraints, and thus relaxations can be solved via second-order cone programming. Our work is closely related with the earlier work of Ceria and Soares (Math Program 86:595-614, 1999) as well as recent work by Frangioni and Gentile (Math Program 106:225-236, 2006) and, Akturk et al. (Oper Res Lett 37:187-191, 2009). Finally, we apply our results to develop tight formulations of mixed integer nonlinear programs in which the nonlinear functions are separable and convex and in which indicator variables play an important role. In particular, we present computational results for three applications quadratic facility location, network design with congestion, and portfolio optimization with buy-in thresholds that show the power of the reformulation technique.

关键词： mixed-integer nonlinear programming Perspective functions

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STUDY ON OPTIMIZATION OF OPERATION FOR MULTIPURPOSE BATCH CHEMICAL-PLANTS WITH MULTIPLE PRODUCTION ROUTES

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COMPUTERS & CHEMICAL ENGINEERING 1995年第0期19卷 S101-S106页

作者： WANG, YP ZHENG, GW XU, YE Department of Chemical Engineering Tianjin University Tianjin 300072 China

According to the operation characteristics of such plants, on the basis of Mauderli-Wellons's research, the mathematical models of operation optimization are modified which consider the campaign formation and production plan problem respectively. A new heuristics is advanced to search for the dominant production campaigns. Comparing the heuristics with Mauderli's algorithm, it ensures that parallel items in-phase have the same batch and that each production line operates with optimized batch, therefore the plant has a larger processing rate. This heuristics can effectively relieve the bottlenecks, and avoid the complexity resulting from the nonconvexity of the model. Within the resource alocated, the mixed-integer algorithm is used to maximize the profit.

关键词： MULTIPURPOSE PLANTS BATCH PLANT OPERATION MULTIPLE ROUTES HEURISTICS mixed-integer nonlinear programming

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Heuristic Optimization Model for the Optimal Layout and Pipe Design of Sewer Systems

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WATER RESOURCES MANAGEMENT 2016年第5期30卷 1605-1620页

作者： Steele, Joshua C. Mahoney, Kurt Karovic, Omer Mays, Larry W. Arizona State Univ Sch Sustainable Engn & Built Environm Tempe AZ 85257 USA

The design of urban stormwater systems and sanitary sewer systems consists of solving two problems: generating a layout of the system and the pipe design which includes the crown elevations, slopes and commercial pipe sizes. A heuristic model for determining the optimal (minimum cost) layout and pipe design of a storm sewer network is presented. The hierarchical procedure combines a sewer layout model formulated as a mixed-integer nonlinear programming (MINLP) problem which is solved using the General Algebraic Modeling System (GAMS) and a simulated annealing optimization procedure for the pipe design of a generated layout was developed in Excel. The GAMS and simulated annealing models are interfaced through linkage of Excel and GAMS. The pipe design model is based upon the simulated annealing method to optimize the crown elevations and diameter of pipe segments in a storm sewer network using layouts generated using GAMS. A sample scenario demonstrates that using these methods may allow for significant costs saving while simultaneously reducing the time typically required to design and compare multiple storm sewer networks.

关键词： Sewer systems Optimization Minimum cost design Hierarchical modeling Simulated annealing mixed-integer nonlinear programming

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Monomial-wise optimal separable underestimators for mixed-integer polynomial optimization

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JOURNAL OF GLOBAL OPTIMIZATION 2017年第4期67卷 759-786页

作者： Buchheim, Christoph D'Ambrosio, Claudia TU Dortmund Fak Math Vogelpothsweg 87 D-44227 Dortmund Germany Ecole Polytech LIX CNRS UMR7161 F-91128 Palaiseau France

We introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches in global optimization are based on convex relaxations. Thanks to the fact that only simple bound constraints are present, minimizing the separable underestimator can be done very quickly. The underestimators are computed monomial-wise after the original polynomial has been shifted. We show that such valid underestimators exist and their degree can be bounded when the degree of the polynomial objective function is bounded, too. For the quartic case, all optimal monomial underestimators are determined analytically. We implemented and tested the branch-and-bound algorithm where these underestimators are hardcoded. The comparison against standard global optimization and polynomial optimization solvers clearly shows that our method outperforms the others, the only exception being the binary case where, though, it is still competitive. Moreover, our branch-and-bound approach suffers less in case of dense polynomial objective function, i.e., in case of polynomials having a large number of monomials. This paper is an extended and revised version of the preliminary paper [4].

关键词： Polynomial optimization mixed-integer nonlinear programming Separable underestimators

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An upgraded superstructure-based model for simultaneous synthesis of direct work and heat exchanger networks

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CHEMICAL ENGINEERING RESEARCH & DESIGN 2020年 159卷 377-394页

作者： Zhuang, Yu Zhang, Lei Liu, Linlin Du, Jian Shen, Shengqiang Dalian Univ Technol Sch Chem Engn Inst Chem Proc Syst Engn Dalian 116024 Liaoning Peoples R China Dalian Univ Technol Sch Energy & Power Engn Key Lab Liaoning Prov Desalinat Dalian 116024 Liaoning Peoples R China

The optimal work and heat integration for pressure and temperature manipulation has been brought into sharp focus due to its paramount significance in reaching considerable energy and expenditure savings. This article proposes an upgraded stage-wise superstructure that simultaneously performs heat integration among unclassified hot/cold streams and direct work integration between pre-classified high/low-pressure streams. An efficiently innovative optimization model for the cost-effective synthesis of direct work and heat exchanger networks (WHENs) is established. A key strategy is to implement the identification of stream types (hot/cold). To this end, an improved superstructure-based model is presented for heat exchanger networks synthesis, combined with an optimized selection of multiple pressure-manipulation units in each stage of the WHEN superstructure. Besides, the proposed method can surpass certain limitations in previous studies that the coupling of direct work integration and heat integration is not considered. Our approach can effectively design preferable WHEN configurations and offer an alternative for enhancing thermal and mechanical energy recovery at the lowest total annual cost. Two case studies are conducted to assess the effectiveness of our proposed methodology, in which the better configurations can be obtained with considerable cost savings of 39.2% and 6.8% when compared to the corresponding literature solutions, respectively. (C) 2020 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

关键词： Superstructure Work and heat exchanger networks Simultaneous synthesis Direct work integration mixed-integer nonlinear programming

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Convex Relaxations for Quadratic On/Off Constraints and Applications to Optimal Transmission Switching

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INFORMS JOURNAL ON COMPUTING 2020年第3期32卷 682-696页

作者： Bestuzheva, Ksenia Hijazi, Hassan Coffrin, Carleton Australian Natl Univ Data61 CSIRO Canberra ACT 2601 Australia Zuse Inst Berlin D-14195 Berlin Germany Los Alamos Natl Lab Los Alamos NM 87545 USA

This paper studies mixed-integer nonlinear programs featuring disjunctive constraints and trigonometric functions and presents a strengthened version of the convex quadratic relaxation of the optimal transmission switching problem. We first characterize the convex hull of univariate quadratic on/off constraints in the space of original variables using perspective functions. We then introduce new tight quadratic relaxations for trigonometric functions featuring variables with asymmetrical bounds. These results are used to further tighten recent convex relaxations introduced for the optimal transmission switching problem in power systems. Using the proposed improvements, along with bound propagation, on 23 medium-sized test cases in the PGLib benchmark library with a relaxation gap of more than 1%, we reduce the gap to less than 1% on five instances. The tightened model has promising computational results when compared with state-of-the-art formulations.

关键词： mixed-integer nonlinear programming perspective relaxation on/off constraints optimal transmission switching trigonometric functions

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Bi-objective optimization of a multi-product multi-period three-echelon supply chain problem under uncertain environments: NSGA-II and NRGA

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INFORMATION SCIENCES 2015年第0期292卷 57-74页

作者： Pasandideh, Seyed Hamid Reza Niaki, Seyed Taghi Akhavan Asadi, Kobra Kharazmi Univ Fac Engn Dept Ind Engn Tehran Iran Sharif Univ Technol Dept Ind Engn Tehran Iran

Bi-objective optimization of a multi-product multi-period three-echelon supply-chain-network problem is aimed in this paper. The network consists of manufacturing plants, distribution centers (DCs), and customer nodes. To bring the problem closer to reality, the majority of the parameters in this network including fixed and variable costs, customer demand, available production time, set-up and production times, all are considered stochastic. The goal is to determine the quantities of the products produced by the manufacturing plants in different periods, the number and locations of the warehouses, the quantities of products transported between the supply chain entities, the inventory of products in warehouses and plants, and the shortage of products in periods such that both the expected and the variance of the total cost are minimized. The problem is first formulated into the framework of a single-objective stochastic mixed integer linear programming model. Then, it is reformulated into a bi-objective deterministic mixed-integer nonlinear programming model. To solve the complicated problem, a non-dominated sorting genetic algorithm (NSGA-II) is utilized next. As there is no benchmark available in the literature, another GA-based algorithm called non-dominated ranking genetic algorithm (NRGA) is used to validate the results obtained. In both algorithms, a modified priority-based encoding is proposed. Some numerical illustrations are provided at the end to not only show the applicability of the proposed methodology, but also to select the best method using a t-test along with the simple additive weighting (SAW) method. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Supply chain management Uncertainty mixed-integer nonlinear programming NRGA & NSGA-II SAW

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