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SIAM JOURNAL ON optimization 2019年第2期29卷 1240-1259页

作者： Xu, Luze Lee, Jon Skipper, Daphne Univ Michigan Dept Ind & Operat Engn Ann Arbor MI 48109 USA US Naval Acad Dept Math Annapolis MD 21402 USA

In the context of global optimization of mixed-integer nonlinear optimization formulations, we consider smoothing univariate functions f that satisfy f(0) = 0, f is increasing and concave on [0, vertical bar infinity), f is twice differentiable on all of (0, vertical bar infinity), but f'(0) is undefined or intolerably large. The canonical examples are root functions f(w) := w(p) for 0 < p < 1. We consider the earlier approach of defining a smoothing function g that is identical with f on (delta,+infinity), for some chosen delta > 0, then replacing the part of f on [0, delta] with the unique homogeneous cubic, matching f, f', and f '' at delta. The parameter delta is used to control (i.e., upper bound) the derivative at 0 (which controls it on all of [0,+infinity) when g is concave). Our main results are as follows: (i) we weaken an earlier sufficient condition to give a necessary and sufficient condition for the piecewise function g to be increasing and concave;(ii) we give a general sufficient condition for g' (0) to be decreasing in the smoothing parameter delta, and under the same condition we demonstrate that the worst-case error of g as an estimate of f is increasing in delta;(iii) we give a general sufficient condition for g to underestimate f;(iv) we give a general sufficient condition for g to dominate the simple "shift smoothing" h(w) := f(w + lambda) f(lambda) (lambda > 0) when the parameters delta and lambda are chosen "fairly," i. e., so that g' (0) = h' (0). In doing so, we solve two natural open problems of Lee and Skipper [J. Global Optim., 69 (2017), pp. 677-697] concerning (iii) and (iv) for root functions.

关键词： global optimization mixed-integer nonlinear optimization spatial branch-and-bound concave nondifferentiable smoothing piecewise

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Operational Planning of Distribution Networks Based on Utility Planning Concepts

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IEEE TRANSACTIONS ON POWER SYSTEMS 2019年第3期34卷 2114-2127页

作者： Mansor, Nurulafiqah Nadzirah Levi, Victor Univ Manchester Fac Sci & Engn Sch Elect & Elect Engn Manchester M13 9PL Lancs England

The paper presents the operational planning of medium voltage (11 kV;6.6 kV) distribution networks that is an integral part of the previously developed integrated planning approach based on utility planning concepts. The operational planning problem is decoupled from the investment planning stage and it is further divided into two phases, the quality of supply (QoS) planning and the minimization of operational costs. The QoS planning maximizes the benefits of the regulatory incentive regime by installing and automating switchgear, as well as finding optimal normally open points. It is solved in two ways, by applying genetic algorithm optimization and then the approach based on engineering rules. Operational cost minimization considers costs of O&M, switching, losses, and reliability. It includes security constraints for radially operated networks and allows that the locations of new switchgear from the QoS stage are changed. The simplified version of the model is also intended to be used as a real-time network re-configuration tool. The entire methodology is tested on the 69-bus test system.

关键词： Distribution operations planning quality of supply operations cost minimization mixed-integer nonlinear optimization optimal reconfiguration

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Optimal price zones of electricity markets: a mixed-integer multilevel model and global solution approaches

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optimization METHODS & SOFTWARE 2019年第2期34卷 406-436页

作者： Grimm, Veronika Kleinert, Thomas Liers, Frauke Schmidt, Martin Zoettl, Gregor Friedrich Alexander Univ Erlangen Nurnberg Chair Econ Theory Lange Gasse 20 D-90403 Nurnberg Germany Energie Campus Nurnberg Further Str 250 D-90429 Nurnberg Germany Friedrich Alexander Univ Erlangen Nurnberg Discrete Optimizat Cauerstr 11 D-91058 Erlangen Germany Friedrich Alexander Univ Erlangen Nurnberg Ind Org & Energy Markets Lange Gasse 20 D-90403 Nurnberg Germany

Mathematical modelling of market design issues in liberalized electricity markets often leads to mixed-integer nonlinear multilevel optimization problems for which no general-purpose solvers exist and which are intractable in general. In this work, we consider the problem of splitting a market area into a given number of price zones such that the resulting market design yields welfare-optimal outcomes. This problem leads to a challenging multilevel model that contains a graph-partitioning problem with multi-commodity flow connectivity constraints and nonlinearities due to proper economic modelling. Furthermore, it has highly symmetric solutions. We develop different problem-tailored solution approaches. In particular, we present an extended Karush-Kuhn-Tucker (KKT) transformation approach as well as a generalized Benders approach that both yield globally optimal solutions. These methods, enhanced with techniques such as symmetry breaking and primal heuristics, are evaluated in detail on academic as well as on realistic instances. It turns out that our approaches lead to effective solution methods for the difficult optimization tasks presented here, where the problem-specific generalized Benders approach performs considerably better than the methods based on KKT transformation.

关键词： multilevel optimization mixed-integer nonlinear optimization graph partitioning generalized Benders decomposition electricity market design

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Computing feasible points for binary MINLPs with MPECs

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MATHEMATICAL PROGRAMMING COMPUTATION 2019年第1期11卷 95-118页

作者： Schewe, Lars Schmidt, Martin Friedrich Alexander Univ Erlangen Nurnberg FAU Discrete Optimizat Cauerstr 11 D-91058 Erlangen Germany Energie Campus Nurnberg Further Str 250 D-90429 Nurnberg Germany

Nonconvex mixed-binary nonlinear optimization problems frequently appear in practice and are typically extremely hard to solve. In this paper we discuss a class of primal heuristics that are based on a reformulation of the problem as a mathematical program with equilibrium constraints. We then use different regularization schemes for this class of problems and use an iterative solution procedure for solving series of regularized problems. In the case of success, these procedures result in a feasible solution of the original mixed-binary nonlinear problem. Since we rely on local nonlinear programming solvers the resulting method is fast and we further improve its reliability by additional algorithmic techniques. We show the strength of our method by an extensive computational study on 662 MINLPLib2 instances, where our methods are able to produce feasible solutions for 60% of all instances in at most 10 s.

关键词： mixed-integer nonlinear optimization MINLP MPEC Complementarity constraints Primal heuristic

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GLOBAL optimization OF mixed-integer ODE CONSTRAINED NETWORK PROBLEMS USING THE EXAMPLE OF STATIONARY GAS TRANSPORT

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SIAM JOURNAL ON optimization 2019年第4期29卷 2949-2985页

作者： Habeck, Oliver Pfetsch, Marc E. Ulbrich, Stefan Tech Univ Darmstadt Res Grp Optimizat Dept Math Darmstadt Germany

In this paper we propose a new approach for finding global solutions for a class of mixed-integer nonlinear optimization problems with ordinary differential equation (ODE) constraints on networks. Instead of using a discretize-then-optimize approach, we combine spatial and variable branching with appropriate discretizations of the differential equations to derive relaxations of the original problem. To construct the relaxations we derive sufficient conditions under which appropriate numerical discretization schemes yield lower and upper bounds on the ODE solutions. Moreover, we derive conditions that ensure the convexity or concavity of the obtained under- and overestimators. Thereby, we make use of the underlying network structure, where the solutions of the ODEs only need to be known at a finite number of points, that is, at the junctions of the network. This property enables us to adaptively refine the discretization and relaxation without introducing new variables in the mixed-integer optimization problem. The incorporation into a spatial branch-and-bound process allows us to compute global epsilon-optimal solutions or to decide infeasibility. We prove that this algorithm terminates finitely under some natural assumptions. We then show how this approach works for the example of stationary gas transport and provide some illustrative computational examples.

关键词： mixed-integer nonlinear optimization global optimization gas networks ODEs on graphs

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Identifying Fixations in Gaze Data via Inner Density and optimization

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INFORMS JOURNAL ON COMPUTING 2019年第3期31卷 459-476页

作者： Trapp, Andrew C. Liu, Wen Djamasbi, Soussan Worcester Polytech Inst Robert A Foisie Business Sch Worcester MA 01609 USA Worcester Polytech Inst Data Sci Program Worcester MA 01609 USA

Eye tracking is an increasingly common technology with a variety of practical uses. Eye-tracking data, or gaze data, can be categorized into two main events: fixations represent focused eye movement, indicative of awareness and attention, whereas saccades are higher-velocity movements that occur between fixation events. Common methods to identify fixations in gaze data can lack sensitivity to peripheral points and may misrepresent positional and durational properties of fixations. To address these shortcomings, we introduce the notion of inner density for fixation identification, which concerns both the duration of the fixation and the proximity of its constituent gaze points. Moreover, we demonstrate how to identify fixations in a sequence of gaze data by optimizing for inner density. After decomposing the clustering of a temporal gaze data sequence into successive regions (chunks), we use nonlinear and linear 0-1 optimization formulations to identify the densest fixations within a given data chunk. Our approach is parametrized by a unique constant that adjusts the degree of desired density, allowing decision makers to have fine-tuned control over density during the process. Computational experiments on real data sets demonstrate the efficiency of our approach and its effectiveness in identifying fixations with greater density than existing methods, thereby enabling the refinement of key gaze metrics such as fixation duration and fixation center.

关键词： eye-tracking gaze data fixation identification fixation inner density mixed-integer nonlinear optimization

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THE MULTILINEAR POLYTOPE FOR ACYCLIC HYPERGRAPHS

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SIAM JOURNAL ON optimization 2018年第2期28卷 1049-1076页

作者： Del Pia, Alberto Khajavirad, Aida Univ Wisconsin Dept Ind & Syst Engn Madison WI 53706 USA Univ Wisconsin Wisconsin Inst Discovery Madison WI 53706 USA Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

We consider the multilinear polytope defined as the convex hull of the set of binary points z satisfying a collection of equations of the form z(e) = Pi(zv)(v is an element of e), e is an element of E, where E denotes a family of subsets of {1, . . . , n} of cardinality at least two. Such sets are of fundamental importance in many types of mixed-integer nonlinear optimization problems, such as 0 - 1 polynomial optimization. Utilizing an equivalent hypergraph representation, we study the facial structure of the multilinear polytope in conjunction with the acyclicity degree of the underlying hypergraph. We provide explicit characterizations of the multilinear polytopes corresponding to Berge-acylic and gamma-acyclic hypergraphs. As the multilinear polytope for gamma-acyclic hypergraphs may contain exponentially many facets in general, we present a strongly polynomial-time algorithm to solve the separation problem, implying polynomial solvability of the corresponding class of 0 - 1 polynomial optimization problems. As an important byproduct, we present a new class of cutting planes for constructing tighter polyhedral relaxations of mixed-integer nonlinear optimization problems with multilinear subexpressions.

关键词： multilinear polytope cutting planes hypergraph acyclicity separation algorithm mixed-integer nonlinear optimization

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Multiobjective Dynamic VAR Planning Strategy With Different Shunt Compensation Technologies

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IEEE TRANSACTIONS ON POWER SYSTEMS 2018年第3期33卷 2429-2439页

作者： Tahboub, Ahmad M. El Moursi, Mohamed Shawky Woon, Wei Lee Kirtley, James L. Khalifa Univ Sci & Technol Masdar Inst Elect Engn & Comp Sci Dept Abu Dhabi 54224 U Arab Emirates UAE Minist Energy Dubai 99979 U Arab Emirates MIT 77 Massachusetts Ave Cambridge MA 02139 USA

High concentrations of induction motor loads can impose stress on transmission and distribution systems, leading to voltage instability in some situations. Properly sized and coordinated reactive power sources will provide for improved operation. We present a strategy for finding an optimal mix (type, size, and location) of dynamic shunt reactive compensation devices. The planning strategy is subject to satisfying steady-state, dynamic and transient performance criteria such as fault-induced delayed voltage recovery limits, as well as criteria related to single (N-1) contingency and load disturbance events. Shunt reactive power compensation devices considered include mechanically switched capacitor banks, static reactive power compensators and static synchronous compensators. The proposed strategy employs a large number of multitimescale time-domain simulations suitable for use with high performance computing clusters and a genetic algorithm to solve the mixed-integer nonlinear programming formulation using parallel computation capabilities. The method is applied to a New England IEEE 39-bus system with assumed high penetration of induction motors. A comprehensive study shows that performance enhancement and significant cost reduction can be achieved using an optimum combination of various shunt compensator technologies.

关键词： Dynamic VAR planning fault-induced delayed voltage recovery genetic algorithm high-performance computing mixed-integer nonlinear optimization multi-objective optimization

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Algorithmic and modeling insights via volumetric comparison of polyhedral relaxations

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MATHEMATICAL PROGRAMMING 2018年第1期170卷 121-140页

作者： Lee, Jon Skipper, Daphne Speakman, Emily Univ Michigan Dept IOE Ann Arbor MI 48109 USA US Naval Acad Dept Math Annapolis MD 21402 USA Otto von Guericke Univ Inst Math Optimizat Magdeburg Germany

This is mostly a survey on some mathematical results concerning volumes of polytopes of interest in non-convex optimization. Our motivation is in geometrically comparing relaxations in the context of mixed-integer linear and nonlinear optimization, with the goal of gaining algorithmic and modeling insights. We consider relaxations of: fixed-charge formulations, vertex packing polytopes, boolean-quadric polytopes, and relaxations of graphs of monomials on box domains. Besides surveying the area, we do give a few new results, and we provide many directions for further work.

关键词： Polytope Volume Global optimization mixed-integer nonlinear optimization Fixed charge Facility location Vertex packing Boolean quadric Monomial Spatial branch-and-bound

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PENALTY ALTERNATING DIRECTION METHODS FOR mixed-integer optimization: A NEW VIEW ON FEASIBILITY PUMPS

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SIAM JOURNAL ON optimization 2017年第3期27卷 1611-1636页

作者： Geissler, Bjorn Morsi, Antonio Schewe, Lars Schmid, Martin Friedrich Alexander Univ Erlangen Nurnberg FAU Discrete Optimizat Cauerstr 11 D-91058 Erlangen Germany Energie Campus Nurnberg Further Str 250 D-90429 Nurnberg Germany

Feasibility pumps are highly effective primal heuristics for mixed-integer linear and nonlinear optimization. However, despite their success in practice there are only a few works considering their theoretical properties. We show that feasibility pumps can be seen as alternating direction methods applied to special reformulations of the original problem, inheriting the convergence theory of these methods. Moreover, we propose a novel penalty framework that encompasses this alternating direction method, which allows us to refrain from random perturbations that are applied in standard versions of feasibility pumps in case of failure. We present a convergence theory for the new penalty based alternating direction method and compare the new variant of the feasibility pump with existing versions in an extensive numerical study for mixed-integer linear and nonlinear problems.

关键词： mixed-integer nonlinear optimization mixed-integer linear optimization feasibility pump alternating direction methods penalty methods

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