Abstract: Abstract Solving 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. PubDate: 2021-02-22

Abstract: Abstract We present two bilevel programming formulations for the aircraft deconfliction problem: one based on speed regulation in k dimensions, the other on heading angle changes in 2 dimensions. We propose three reformulations of each problem based on KKT conditions and on two different duals of the lower-level subproblems. We also propose a cut generation algorithm to solve the bilevel formulations. Finally, we present computational results on a variety of instances. PubDate: 2021-02-18

Abstract: Abstract The paper is dedicated to the study of strong duality for a problem of linear copositive programming. Based on the recently introduced concept of the set of normalized immobile indices, an extended dual problem is deduced. The dual problem satisfies the strong duality relations and does not require any additional regularity assumptions such as constraint qualifications. The main difference with the previously obtained results consists in the fact that now the extended dual problem uses neither the immobile indices themselves nor the explicit information about the convex hull of these indices. The strong duality formulations presented in the paper for linear copositive problems have similar structure and properties as that proposed in the works by M. Ramana, L. Tuncel, and H. Wolkowicz, for semidefinite programming. PubDate: 2021-02-18

Abstract: Abstract Relaxed correlation clustering (RCC) is a vertex partitioning problem that aims at minimizing the so-called relaxed imbalance in signed graphs. RCC is considered to be an NP-hard unsupervised learning problem with applications in biology, economy, image recognition and social network analysis. In order to solve it, we propose two linear integer programming formulations and a local search-based metaheuristic. The latter relies on auxiliary data structures to efficiently perform move evaluations during the search process. Extensive computational experiments on existing and newly proposed benchmark instances demonstrate the superior performance of the proposed approaches when compared to those available in the literature. While the exact approaches obtained optimal solutions for open problems, the proposed heuristic algorithm was capable of finding high quality solutions within a reasonable CPU time. In addition, we also report improving results for the symmetrical version of the problem. Moreover, we show the benefits of implementing the efficient move evaluation procedure that enables the proposed metaheuristic to be scalable, even for large-size instances. PubDate: 2021-02-13

Abstract: Abstract In this paper, a power penalty approximation method is proposed for solving a mixed quasilinear elliptic complementarity problem. The mixed complementarity problem is first reformulated as a double obstacle quasilinear elliptic variational inequality problem. A nonlinear elliptic partial differential equation is then defined to approximate the resulting variational inequality by using a power penalty approach. The existence and uniqueness of the solution to the partial differential penalty equation are proved. It is shown that, under some mild assumptions, the sequence of solutions to the penalty equations converges to the unique solution of the variational inequality problem as the penalty parameter tends to infinity. The error estimates of the convergence of this penalty approach are also derived. At last, numerical experimental results are presented to show that the power penalty approximation method is efficient and robust. PubDate: 2021-02-12

Abstract: Abstract The max-k-cut problem is one of the most well-known combinatorial optimization problems. In this paper, we design an efficient branch-and-bound algorithm to solve the max-k-cut problem. We propose a semidefinite relaxation that is as tight as the conventional semidefinite relaxations in the literature, but is more suitable as a relaxation method in a branch-and-bound framework. We then develop a branch-and-bound algorithm that exploits the unique structure of the proposed semidefinite relaxation, and applies a branching method different from the one commonly used in the existing algorithms. The symmetric structure of the solution set of the max-k-cut problem is discussed, and a strategy is devised to reduce the redundancy of subproblems in the enumeration procedure. The numerical results show that the proposed algorithm is promising. It performs better than Gurobi on instances that have more than 350 edges, and is more efficient than the state-of-the-art algorithm bundleBC on certain types of test instances. PubDate: 2021-02-11

Abstract: Abstract The emphasis of the paper is to examine the essential stability of efficient solutions for semicontinuous vector optimization problems, subject to the perturbation of objective function and feasible set. We obtain sufficient conditions for existence and characterization of essential efficient solutions, essential sets and essential components, where the efficient solutions are governed by an arbitrary preference relation in a real normed linear space. Further, we establish the density of the set of stable vector optimization problems in the sense of Baire category. We also exhibit that essential stability is weaker than examining continuity aspects of solution sets. PubDate: 2021-02-09

Abstract: Abstract The computation of lower bounds via the solution of convex lower bounding problems depicts current state-of-the-art in deterministic global optimization. Typically, the nonlinear convex relaxations are further underestimated through linearizations of the convex underestimators at one or several points resulting in a lower bounding linear optimization problem. The selection of linearization points substantially affects the tightness of the lower bounding linear problem. Established methods for the computation of such linearization points, e.g., the sandwich algorithm, are already available for the auxiliary variable method used in state-of-the-art deterministic global optimization solvers. In contrast, no such methods have been proposed for the (multivariate) McCormick relaxations. The difficulty of determining a good set of linearization points for the McCormick technique lies in the fact that no auxiliary variables are introduced and thus, the linearization points have to be determined in the space of original optimization variables. We propose algorithms for the computation of linearization points for convex relaxations constructed via the (multivariate) McCormick theorems. We discuss alternative approaches based on an adaptation of Kelley’s algorithm; computation of all vertices of an n-simplex; a combination of the two; and random selection. All algorithms provide substantial speed ups when compared to the single point strategy used in our previous works. Moreover, we provide first results on the hybridization of the auxiliary variable method with the McCormick technique benefiting from the presented linearization strategies resulting in additional computational advantages. PubDate: 2021-02-09

Abstract: Abstract This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness. PubDate: 2021-02-08

Abstract: Abstract In this work, we propose a segment-based representation for the upper bound of the non-dominated set in interval branch & bound solvers for biobjective non linear optimization. We ensure that every point over the upper line segments is dominated by at least one point in the feasible objective region. Segments are generated by linear envelopes of the image of feasible line segments. Finally, we show that the segment-based representation together with methods for generating upper line segments allows us to converge more quickly to the desired precision of the whole strategy. The code of our solver can be found in our git repository (https://github.com/INFPUCV/ibex-lib/tree/master/plugins/optim-mop). PubDate: 2021-02-05

Abstract: Abstract This paper addresses an extension of the flexible job shop scheduling problem by considering that jobs need to be moved around the shop-floor by a set of vehicles. Thus, this problem involves assigning each production operation to one of the alternative machines, finding the sequence of operations for each machine, assigning each transport task to one of the vehicles, and finding the sequence of transport tasks for each vehicle, simultaneously. Transportation is usually neglected in the literature and when considered, an unlimited number of vehicles is, typically, assumed. Here, we propose the first mixed integer linear programming model for this problem and show its efficiency at solving small-sized instances to optimality. In addition, and due to the NP-hard nature of the problem, we propose a local search based heuristic that the computational experiments show to be effective, efficient, and robust. PubDate: 2021-02-05

Abstract: Abstract The capacitated vehicle routing problem (CVRP) is the well-known combinatorial optimization problem having numerous practically important applications. CVRP is strongly NP-hard (even on the Euclidean plane), hard to approximate in general case and APX-complete for an arbitrary metric. Meanwhile, for the geometric settings of the problem, there are known a number of quasi-polynomial and even polynomial time approximation schemes. Among these results, the well-known QPTAS proposed by Das and Mathieu appears to be the most general. In this paper, we propose the first extension of this scheme to a more wide class of metric spaces. Actually, we show that the metric CVRP has a QPTAS any time when the problem is set up in the metric space of any fixed doubling dimension \(d>1\) and the capacity does not exceed \(\mathrm {polylog}{(n)}\) . PubDate: 2021-01-29

Abstract: Abstract We develop convexification techniques for mathematical programs with complementarity constraints. Specifically, we adapt the reformulation-linearization technique of Sherali and Adams (SIAM J Discrete Math 3, 411–430, 1990) to problems with linear complementarity constraints and discuss how this procedure reduces to its standard specification for binary mixed-integer programs. Then, we consider specially structured complementarity sets that appear in KKT systems with linear constraints. For sets with a single complementarity constraint, we develop a convexification procedure that generates all nontrivial facet-defining inequalities and has an appealing “cancel-and-relax” interpretation. This procedure is used to describe the convex hull of problems with few side constraints in closed-form. As a consequence, we delineate cases when the factorable relaxation techniques yield the convex hull from those for which they do not. We then discuss how these results extend to sets with multiple complementarity constraints. In particular, we show that a suitable sequential application of the cancel-and-relax procedure produces all nontrivial inequalities of their convex hull. We conclude by illustrating, on a set of randomly generated problems, that the relaxations we propose can be significantly stronger than those available in the literature. PubDate: 2021-01-27

Abstract: Abstract In this paper, we derive (partial) convex hull for deterministic multi-constraint polyhedral conic mixed integer sets with multiple integer variables using conic mixed integer rounding (CMIR) cut-generation procedure of Atamtürk and Narayanan (Math Prog 122:1–20, 2008), thereby extending their result for a simple polyhedral conic mixed integer set with single constraint and one integer variable. We then introduce two-stage stochastic p-order conic mixed integer programs (denoted by TSS-CMIPs) in which the second stage problems have sum of \(l_p\) -norms in the objective function along with integer variables. First, we present sufficient conditions under which the addition of scenario-based nonlinear cuts in the extensive formulation of TSS-CMIPs is sufficient to relax the integrality restrictions on the second stage integer variables without impacting the integrality of the optimal solution of the TSS-CMIP. We utilize scenario-based CMIR cuts for TSS-CMIPs and their distributionally robust generalizations with structured CMIPs in the second stage, and prove that these cuts provide conic/linear programming equivalent or approximation for the second stage CMIPs. We also perform extensive computational experiments by solving stochastic and distributionally robust capacitated facility location problem and randomly generated structured TSS-CMIPs with polyhedral CMIPs and second-order CMIPs in the second stage, i.e. \(p=1\) and \(p =2\) , respectively. We observe that there is a significant reduction in the total time taken to solve these problems after adding the scenario-based cuts. PubDate: 2021-01-22

Abstract: Abstract Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples. PubDate: 2021-01-19

Abstract: Abstract In this paper, we extend the purely dual formulation that we recently proposed for the three-dimensional assignment problems to solve the more general multidimensional assignment problem. The convex dual representation is derived and its relationship to the Lagrangian relaxation method that is usually used to solve multidimensional assignment problems is investigated. Also, we discuss the condition under which the duality gap is zero. It is also pointed out that the process of Lagrangian relaxation is essentially equivalent to one of relaxing the binary constraint condition, thus necessitating the auction search operation to recover the binary constraint. Furthermore, a numerical algorithm based on the dual formulation along with a local search strategy is presented. The simulation results show that the proposed algorithm outperforms the traditional Lagrangian relaxation approach in terms of both accuracy and computational efficiency. PubDate: 2021-01-18

Abstract: Abstract Recommender systems make use of different sources of information for providing users with recommendations of items. Such systems are often based on either collaborative filtering methods which make automatic predictions about the interests of a user, using preferences of similar users, or content based filtering that matches the user’s personal preferences with item characteristics. We adopt the content-based approach and propose to use the concept of resolving set that allows to determine the preferences of the users with a very limited number of ratings. We also show how to make recommendations when user ratings are imprecise or inconsistent, and we indicate how to take into account situations where users possibly don’t care about the attribute values of some items. All recommendations are obtained in a few seconds by solving integer programs. PubDate: 2021-01-09

Abstract: Abstract We propose a convex quadratic programming (CQP) relaxation for multi-ball constrained quadratic optimization (MB). (CQP) is shown to be equivalent to semidefinite programming relaxation in the hard case. Based on (CQP), we propose an algorithm for solving (MB), which returns a solution of (MB) with an approximation bound independent of the number of constraints. The approximation algorithm is further extended to solve nonconvex quadratic optimization with more general constraints. As an application, we propose a standard quadratic programming relaxation for finding Chebyshev center of a general convex set with a guaranteed approximation bound. PubDate: 2021-01-09