Nonconvex Programming Research Articles

Nonlinear rescaling Lagrangians for nonconvex semidefinite programming

This paper focuses on the study of rescaling Lagrangians for solving nonconvex semidefinite programming problems. The rescaling nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions is proposed to guarantee the convergence of nonlinear rescaling Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence theorem shows that, under the second-order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the nonlinear rescaling Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in the nonlinear rescaling Lagrangian method for nonlinear programming, we have to deal with the sigma term in the convergence analysis.

A parametric solution algorithm for a class of rank-two nonconvex programs

The aim of this paper is to propose a solution algorithm for a particular class of rank-two nonconvex programs having a polyhedral feasible region. The algorithm lies within the class of the so called "optimal level solutions" parametric methods. The subproblems obtained by means of this parametrical approach are quadratic convex ones, but not necessarily neither strictly convex nor linear. For this very reason, in order to solve in an unifying framework all of the considered rank-two nonconvex programs a new approach needs to be proposed. The efficiency of the algorithm is improved by means of the use of underestimation functions. The results of a computational test are provided and discussed.

Optimizing a multi-stage production/inventory system by DC programming based approaches

This paper deals with optimizing the cost of set up, transportation and inventory of a multi-stage production system in presence of bottleneck. The considered optimization model is a mixed integer nonlinear program. We propose two methods based on DC (Difference of Convex) programming and DCA (DC Algorithm)--an innovative approach in nonconvex programming framework. The mixed integer nonlinear problem is first reformulated as a DC program and then DCA is developed to solve the resulting problem. In order to globally solve the problem, we combine DCA with a Branch and Bound algorithm (BB-DCA). A convex minorant of the objective function is introduced. DCA is used to compute upper bounds while lower bounds are calculated from a convex relaxation problem. The numerical results compared with those of COUENNE ( http://www.coin-or.org/download/binary/Couenne/ ), a solver for mixed integer nonconvex programming, show the rapidity and the ∈-globality of DCA in almost cases, as well as the efficiency of the combined DCA-Branch and Bound algorithm. We also propose a simple heuristic algorithm which is proved by experimental results to be better than an existing heuristic in the literature for this problem.

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi: 10.1007/s10898-012-9874-7 ), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers.

Extended radial epiderivatives of non-convex vector-valued maps and parametric quasiconvex programming

In this paper, we consider extended vector-valued mappings defined on a normed linear space. Based on the recent semicontinuous regularizations related to hypographical and/or epigraphical profile mappings of the considered function introduced, we define semicontinuous radial epiderivatives. We, then, demonstrate that the properties of these epiderivatives amount to properties of hypographical and/or epigraphical profile mappings of the corresponding difference quotient of the underlying function, which simplify fairly well the proofs in the radial epiderivative formulaes. In particular, we stress the impact of semicontinuity, hence, we characterize with new arguments the radial epiderivatives in terms of the suprema and/or infima of the interiorly radial cone of the hypograph and/or epigraph of the considered function. Finally, we obtain optimality conditions for general non-convex constrained vector optimization problems. We apply thereafter the obtained pattern to a parametric quasiconvex programming problem for which we derive necessary and sufficient optimality conditions that are not sensitive to perturbation at the nominal level, yielding henceforth more – and strong at least under asymptotically regular constraints – information than the recent stability results obtained under additional conditions on the regularity of the normal cone to the adjusted sublevel sets of the underlying function.

Liner shipping network design with deadlines

It is crucial for a liner shipping company to design its container shipping network. Given a set of port-to-port container shipment demands with delivery deadlines, the liner shipping company aims to design itineraries of portcalls, deploy ships on these itineraries and determine how to transport containers with the deployed ships in order to maximize its total profit. In this paper we first demonstrate NP-hardness of this problem and subsequently formulate it as a mixed-integer non-linear non-convex programming model. A column generation based heuristic method is proposed for solving this problem. Numerical experiments for container shipping on the Asia–Europe trade lane show that the proposed solution algorithm is efficient to find good quality solutions.

Communication Network Design: Balancing Modularity and Mixing via Optimal Graph Spectra

By leveraging information technologies, organizations now have the ability to design their communication networks and crowdsourcing platforms to pursue various performance goals, but existing research on network design does not account for the specific features of social networks, such as the notion of teams. We fill this gap by demonstrating how desirable aspects of organizational structure can be mapped parsimoniously onto the spectrum of the graph Laplacian allowing the specification of structural objectives and build on recent advances in non-convex programming to optimize them. This design framework is general, but we focus here on the problem of creating graphs that balance high modularity and low mixing time, and show how liaisons rather than brokers maximize this objective.

New and efficient algorithms for transfer prices and inventory holding policies in two-enterprise supply chains

We consider a multi-period problem of fair transfer prices and inventory holding policies in two enterprise supply chains. This problem was formulated as a mixed integer non-linear program by Gjerdrum et al. (Eur J Oper Res 143:582---599, 2002). Existing global optimization methods to solve this problem are computationally expensive. We propose a continuous approach based on difference of convex functions (DC) programming and DC Algorithms (DCA) for solving this combinatorial optimization problem. The problem is first reformulated as a DC program via an exact penalty technique. Afterward, DCA, an efficient local approach in non-convex programming framework, is investigated to solve the resulting problem. For globally solving this problem, we investigate a combined DCA-Branch and Bound algorithm. DCA is applied to get lower bounds while upper bounds are computed from a relaxation problem. The numerical results on several test problems show that the proposed algorithms are efficient: DCA provides a good integer solution in a short CPU time although it works on a continuous domain, and the combined DCA-Branch and Bound finds an $$\epsilon $$ ∈ -optimal solution for large-scale problems in a reasonable time.

Optimality conditions for minimizing the difference of nonconvex vector-valued mappings

In this paper, we consider vector optimization problems involving the difference of nonconvex vector-valued mappings. By a nonconvex scalarization function, we establish necessary optimality conditions in terms of the Mordukhovich subdifferential, strong subdifferential and Ioffe subdifferential without any convexity assumption. As an application, we discuss the optimality condition on a nonconvex multiobjective fractional programming problem.

Coupled stochastic soil moisture simulation‐optimization model of deficit irrigation

This study presents an explicit stochastic optimization‐simulation model of short‐term deficit irrigation management for large‐scale irrigation districts. The model which is a nonlinear nonconvex program with an economic objective function is built on an agrohydrological simulation component. The simulation component integrates (1) an explicit stochastic model of soil moisture dynamics of the crop‐root zone considering interaction of stochastic rainfall and irrigation with shallow water table effects, (2) a conceptual root zone salt balance model, and 3) the FAO crop yield model. Particle Swarm Optimization algorithm, linked to the simulation component, solves the resulting nonconvex program with a significantly better computational performance compared to a Monte Carlo‐based implicit stochastic optimization model. The model has been tested first by applying it in single‐crop irrigation problems through which the effects of the severity of water deficit on the objective function (net benefit), root‐zone water balance, and irrigation water needs have been assessed. Then, the model has been applied in Dasht‐e‐Abbas and Ein‐khosh Fakkeh Irrigation Districts (DAID and EFID) of the Karkheh Basin in southwest of Iran. While the maximum net benefit has been obtained for a stress‐avoidance (SA) irrigation policy, the highest water profitability has been resulted when only about 60% of the water used in the SA policy is applied. The DAID with respectively 33% of total cultivated area and 37% of total applied water has produced only 14% of the total net benefit due to low‐valued crops and adverse soil and shallow water table conditions.

Joint beamforming design and base‐station assignment in a coordinated multicell system

This study is concerned with the downlink beamforming designs in a coordinated multicell system with dynamic base-station (BS) assignment. At each cell, a multiple-antenna BS employs linear beamforming to send multiple data streams to its assigned mobile-stations (MSs). Exploiting multicell coordination, the multiple BSs jointly optimise the beamformers and the BS-MS assignments to enhance the overall system performance. With per-BS power constraints, considered are the coordinated beamforming problems under the following two design criteria: (i) minimising the transmit power margin at the BS with a set of target signal-to-interference-plus-noise ratios (SINR) at the MSs and (ii) jointly maximising the minimum SINR margin at the MSs. As the original problem formulations are shown to be non-convex integer programs, which are combinatorially hard, the authors propose an efficient convex relaxation approach to solve the problems with low complexity. Simulations show that the convex relaxation-based assignment schemes significantly outperform heuristic fixed assignment schemes.

Approximating the Lovász θ Function with the Subgradient Method

Abstract The famous Lovasz theta number θ ( G ) is expressed as the optimal solution of a semidefinite program. As such, it can be computed in polynomial time to an arbitrary precision. Nevertheless, computing it in practice yields some difficulties as the size of the graph gets larger and larger, despite recent significant advances of semidefinite programming (SDP) solvers. We present a way around SDP which exploits a well-known equivalence between SDP and lagrangian relaxations of non-convex quadratic programs. This allows us to design a subgradient algorithm which is shown to be competitive with SDP algorithms in terms of efficiency, while being preferable as far as memory requirements, flexibility and stability are concerned.

Iterative D.C. Optimization of Precoding in Wireless MIMO Relaying

Optimizations of precoding matrices in precode-and-forward (PF) MIMO relaying are nonconvex programs in precoding matrix variables. The semidefinite relaxation (SDR) technique, which relaxes the concerned nonconvex quadratic constraints by (convex) semi-definite ones, can locate the optimal solutions, provided that the numbers of relaying antennas and users are very small. The computational complexity of the SDR grows explosively even with a very moderate increase in the numbers of relaying antennas and/or users, making the existing semidefinite programming (SDP) solvers incapable. In this paper, much more efficient problem formulations of precoding matrix design that exploit the spectral matrix optimization are developed. Such formulations have a low dimensionality and are computationally-tractable nonconvex matrix programs. Furthermore, by exploiting their partial convex structures in the d.c. (difference of two convex functions) framework, new effective iterative solutions are obtained. Extensive simulation results are presented to support the computational advantage of the proposed approach and show that the proposed approach can effectively handle all three considered optimization problems of precoding matrices in MIMO PF relaying, while the SDR approach either is computationally impractical or fails.

Solving the canonical dual of box- and integer-constrained nonconvex quadratic programs via a deterministic direct search algorithm

This paper presents a massively parallel global deterministic direct search method (VTDIRECT) for solving nonconvex quadratic minimization problems with either box or±1 integer constraints. Using the canonical dual transformation, these well-known NP-hard problems can be reformulated as perfect dual stationary problems (with zero duality gap). Under certain conditions, these dual problems are equivalent to smooth concave maximization over a convex feasible space. Based on a perturbation method proposed by Gao, the integer programming problem is shown to be equivalent to a continuous unconstrained Lipschitzian global optimization problem. The parallel algorithm VTDIRECT is then applied to solve these dual problems to obtain global minimizers. Parallel performance results for several nonconvex quadratic integer programming problems are reported.

Truncated aggregate homotopy method for nonconvex nonlinear programming

In this paper, an efficient implementation of aggregate homotopy method for nonconvex nonlinear programming problems is proposed. Adopting truncated aggregate technique, only a small subset of the constraints is aggregated at each iteration, hence the number of gradient and Hessian calculations is reduced dramatically. The subset is adaptively updated with some cheaply implementable truncating criterions, to guarantee the locally quadratic convergence of the correction process with as few computation cost as possible. Numerical tests with comparison to some other methods show that the new method is very efficient, especially for the problems with large amount of constraints and {computationally expensive} gradients or Hessians.

Dual-based bounds for resource allocation in zero-forcing beamforming OFDMA-SDMA systems

We consider multi-antenna base stations using orthogonal frequency division multiple access and space division multiple access techniques to serve single-antenna users. Some users, called real-time users, have minimum rate requirements and must be served in the current time slot while others, called non real-time users, do not have strict timing constraints and are served on a best-effort basis. The resource allocation problem is to find the assignment of users to subcarriers and the transmit beamforming vectors that maximize the total user rates subject to power and minimum rate constraints. In general, this is a nonlinear and non-convex program and the zero-forcing technique used here makes it integer as well, exact optimal solutions cannot be computed in reasonable time for realistic cases. For this reason, we present a technique to compute both upper and lower bounds and show that these are quite close for some realistic cases. First, we formulate the dual problem whose optimum provides an upper bound to all feasible solutions. We then use a simple method to get a primal-feasible point starting from the dual optimal solution, which is a lower bound on the primal optimal solution. Numerical results for several cases show that the two bounds are close so that the dual method can be used to benchmark any heuristic used to solve this problem. As an example, we provide numerical results showing the performance gap of the well-known weight adjustment method and show that there is considerable room for improvement.

Convexification method for bilinear transmission expansion problem

SUMMARY This research presents a method for convexifying the non-linear constraints and the objective function for the Transmission Network Expansion Planning Problem (TNEP). The TNEP seeks to identify the best set of transmission capacity additions to meet a future electric power demand. The TNEP is a non-convex Mixed Integer Non-linear Programming problem for which a variety of primal solution methods have been designed. In this paper, the TNEP is formulated as a bilinear programming problem subject to bilinear constraints. This allows resolving the non-linearities through convexification of the non-linear objective function and the constraints for a relaxation of the TNEP in which integrality requirements are removed. The final solution for the original TNEP problem is obtained by using a branch and bound strategy. It was found by using a two-node test case that the convexification method presented in this paper in conjunction with a brand and bound strategy identifies the optimal solution. Successfully solving a small-scale test problem through convexification presents a promising scenario to solve larger scale problems. This primal-dual method allows identifying the duality gap which is a limitation of primal methods. Further multidisciplinary research is needed to extend the method presented here to larger cases and real life networks. Copyright © 2013 John Wiley & Sons, Ltd.

A Global Optimization Algorithm for Generalized Quadratic Programming

We present a global optimization algorithm for solving generalized quadratic programming (GQP), that is, nonconvex quadratic programming with nonconvex quadratic constraints. By utilizing a new linearizing technique, the initial nonconvex programming problem (GQP) is reduced to a sequence of relaxation linear programming problems. To improve the computational efficiency of the algorithm, a range reduction technique is employed in the branch and bound procedure. The proposed algorithm is convergent to the global minimum of the (GQP) by means of the subsequent solutions of a series of relaxation linear programming problems. Finally, numerical results show the robustness and effectiveness of the proposed algorithm.

Contribution of copositive formulations to the graph partitioning problem

This article provides analysis of several copositive formulations of the graph partitioning problem and semidefinite relaxations based on them. We prove that the copositive formulations based on results from Burer [S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120 (Ser. A) (2009), pp. 479–495] and the author of the paper [J. Povh, Semidefinite approximations for quadratic programs over orthogonal matrices. J. Global Optim. 48 (2010), pp. 447–463] are equivalent and that they both imply semidefinite relaxations which are stronger than the Donath–Hoffman eigenvalue lower bound [W.E. Donath and A.J. Hoffman, Lower bounds for the partitioning of graphs. IBM J. Res. Develop. 17 (1973), pp. 420–425] and the projected semidefinite lower bound from Wolkowicz and Zhao [H. Wolkowicz and Q. Zhao, Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math. 96–97 (1999), pp. 461–479].

A structured trust region method for nonconvex programming with separable structure

In this paper, we present a structured trust region algorithm for nonconvex programming with separable structure. We obtain the trial step by decomposing the step into its normal and tangential components. The structure of the problem is dealt with in the framework of the trust region. The global convergence is proved for the proposed algorithm. The preliminary numerical results show that the proposed algorithm is potentially efficient.