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Showing papers in "Computational Optimization and Applications in 2018"


Journal ArticleDOI
TL;DR: A proximal difference-of-convex algorithm with extrapolation to possibly accelerate the proximal DCA, and it is shown that any cluster point of the sequence generated by the algorithm is a stationary points of the DC optimization problem for a fairly general choice of extrapolation parameters.
Abstract: We consider a class of difference-of-convex (DC) optimization problems whose objective is level-bounded and is the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a continuous concave function. While this kind of problems can be solved by the classical difference-of-convex algorithm (DCA) (Pham et al. Acta Math Vietnam 22:289–355, 1997), the difficulty of the subproblems of this algorithm depends heavily on the choice of DC decomposition. Simpler subproblems can be obtained by using a specific DC decomposition described in Pham et al. (SIAM J Optim 8:476–505, 1998). This decomposition has been proposed in numerous work such as Gotoh et al. (DC formulations and algorithms for sparse optimization problems, 2017), and we refer to the resulting DCA as the proximal DCA. Although the subproblems are simpler, the proximal DCA is the same as the proximal gradient algorithm when the concave part of the objective is void, and hence is potentially slow in practice. In this paper, motivated by the extrapolation techniques for accelerating the proximal gradient algorithm in the convex settings, we consider a proximal difference-of-convex algorithm with extrapolation to possibly accelerate the proximal DCA. We show that any cluster point of the sequence generated by our algorithm is a stationary point of the DC optimization problem for a fairly general choice of extrapolation parameters: in particular, the parameters can be chosen as in FISTA with fixed restart (O’Donoghue and Candes in Found Comput Math 15, 715–732, 2015). In addition, by assuming the Kurdyka-Łojasiewicz property of the objective and the differentiability of the concave part, we establish global convergence of the sequence generated by our algorithm and analyze its convergence rate. Our numerical experiments on two difference-of-convex regularized least squares models show that our algorithm usually outperforms the proximal DCA and the general iterative shrinkage and thresholding algorithm proposed in Gong et al. (A general iterative shrinkage and thresholding algorithm for non-convex regularized optimization problems, 2013).

112 citations


Journal ArticleDOI
TL;DR: A generalized symmetric ADMM, which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming and is more flexible and attractive on choosing and using larger stepsizes of the dual variable.
Abstract: The alternating direction method of multipliers (ADMM) has been proved to be effective for solving separable convex optimization subject to linear constraints. In this paper, we propose a generalized symmetric ADMM (GS-ADMM), which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming. This GS-ADMM partitions the data into two group variables so that one group consists of p block variables while the other has q block variables, where $$p \ge 1$$ and $$q \ge 1$$ are two integers. The two grouped variables are updated in a Gauss–Seidel scheme, while the variables within each group are updated in a Jacobi scheme, which would make it very attractive for a big data setting. By adding proper proximal terms to the subproblems, we specify the domain of the stepsizes to guarantee that GS-ADMM is globally convergent with a worst-case $${\mathcal {O}}(1/t)$$ ergodic convergence rate. It turns out that our convergence domain of the stepsizes is significantly larger than other convergence domains in the literature. Hence, the GS-ADMM is more flexible and attractive on choosing and using larger stepsizes of the dual variable. Besides, two special cases of GS-ADMM, which allows using zero penalty terms, are also discussed and analyzed. Compared with several state-of-the-art methods, preliminary numerical experiments on solving a sparse matrix minimization problem in the statistical learning show that our proposed method is effective and promising.

53 citations


Journal ArticleDOI
TL;DR: In this article, a two-stage stochastic submodular optimization model is proposed to solve the influence maximization problem with a finite number of samples, where the submodularity of the influence function can be exploited to develop strong optimality cuts.
Abstract: We consider stochastic influence maximization problems arising in social networks. In contrast to existing studies that involve greedy approximation algorithms with a 63% performance guarantee, our work focuses on solving the problem optimally. To this end, we introduce a new class of problems that we refer to as two-stage stochastic submodular optimization models. We propose a delayed constraint generation algorithm to find the optimal solution to this class of problems with a finite number of samples. The influence maximization problems of interest are special cases of this general problem class. We show that the submodularity of the influence function can be exploited to develop strong optimality cuts that are more effective than the standard optimality cuts available in the literature. Finally, we report our computational experiments with large-scale real-world datasets for two fundamental influence maximization problems, independent cascade and linear threshold, and show that our proposed algorithm outperforms the basic greedy algorithm of Kempe et al. (Proceedings of the ninth ACM SIGKDD international conference on knowledge discovery and data mining, KDD'03, New York, NY, USA, ACM, pp 137---146, 2003).

47 citations


Journal ArticleDOI
TL;DR: The developed global convergence theory is stronger than the ones known for current algorithms with convergence to second-order points in the sense that, besides the flexibility introduced by the general lower-level approach, it includes a loose requirement for the resolution of subproblems.
Abstract: Augmented Lagrangian methods with convergence to second-order stationary points in which any constraint can be penalized or carried out to the subproblems are considered in this work. The resolution of each subproblem can be done by any numerical algorithm able to return approximate second-order stationary points. The developed global convergence theory is stronger than the ones known for current algorithms with convergence to second-order points in the sense that, besides the flexibility introduced by the general lower-level approach, it includes a loose requirement for the resolution of subproblems. The proposed approach relies on a weak constraint qualification, that allows Lagrange multipliers to be unbounded at the solution. It is also shown that second-order resolution of subproblems increases the chances of finding a feasible point, in the sense that limit points are second-order stationary for the problem of minimizing the squared infeasibility. The applicability of the proposed method is illustrated in numerical examples with ball-constrained subproblems.

42 citations


Journal ArticleDOI
TL;DR: In this article, the affine policy is reformulated as a copositive optimization problem, which leads to a class of tractable, semidefinite-based approximations.
Abstract: We study two-stage adjustable robust linear programming in which the right-hand sides are uncertain and belong to a convex, compact uncertainty set. This problem is NP-hard, and the affine policy is a popular, tractable approximation. We prove that under standard and simple conditions, the two-stage problem can be reformulated as a copositive optimization problem, which in turn leads to a class of tractable, semidefinite-based approximations that are at least as strong as the affine policy. We investigate several examples from the literature demonstrating that our tractable approximations significantly improve the affine policy. In particular, our approach solves exactly in polynomial time a class of instances of increasing size for which the affine policy admits an arbitrarily large gap.

41 citations


Journal ArticleDOI
TL;DR: An instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph is proposed, which is shown to be well-posed and affine-linear solutions of these systems are derived analytically.
Abstract: We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.

39 citations


Journal ArticleDOI
TL;DR: In this paper, a new algorithm for testing copositivity of high order tensors, and then present applications of the algorithm in physics and hypergraphs, is proposed with the help of a proper convex subcone of the copositive tensor cone.
Abstract: Copositivity of tensors plays an important role in vacuum stability of a general scalar potential, polynomial optimization, tensor complementarity problem and tensor generalized eigenvalue complementarity problem. In this paper, we propose a new algorithm for testing copositivity of high order tensors, and then present applications of the algorithm in physics and hypergraphs. For this purpose, we first give several new conditions for copositivity of tensors based on the representative matrix of a simplex. Then a new algorithm is proposed with the help of a proper convex subcone of the copositive tensor cone, which is defined via the copositivity of Z-tensors. Furthermore, by considering a sum-of-squares program problem, we define two new subsets of the copositive tensor cone and discuss their convexity. As an application of the proposed algorithm, we prove that the coclique number of a uniform hypergraph is equivalent to an optimization problem over the completely positive tensor cone, which implies that the proposed algorithm can be applied to compute an upper bound of the coclique number of a uniform hypergraph. Then we study another application of the proposed algorithm on particle physics in testing copositivity of some potential fields. At last, various numerical examples are given to show the performance of the algorithm.

39 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proposed a way to extend the Nelder-Mead (NM) algorithm to inequality constrained optimization problems through a search step of the mesh adaptive direct search (Mads) algorithm, inspired by the NM algorithm.
Abstract: Despite the lack of theoretical and practical convergence support, the Nelder–Mead (NM) algorithm is widely used to solve unconstrained optimization problems. It is a derivative-free algorithm, that attempts iteratively to replace the worst point of a simplex by a better one. The present paper proposes a way to extend the NM algorithm to inequality constrained optimization. This is done through a search step of the mesh adaptive direct search (Mads) algorithm, inspired by the NM algorithm. The proposed algorithm does not suffer from the NM lack of convergence, but instead inherits from the totality of the Mads convergence analysis. Numerical experiments show an important improvement in the quality of the solutions produced using this search step.

37 citations


Journal ArticleDOI
TL;DR: This paper proposes a class of specific ADMM-based algorithms that can be used for three-block separable convex minimization models and shows the contraction property, prove the global convergence and establish the worst-case convergence rate measured by the iteration complexity for this prototype algorithm.
Abstract: The alternating direction method of multipliers (ADMM) recently has found many applications in various domains whose models can be represented or reformulated as a separable convex minimization model with linear constraints and an objective function in sum of two functions without coupled variables. For more complicated applications that can only be represented by such a multi-block separable convex minimization model whose objective function is the sum of more than two functions without coupled variables, it was recently shown that the direct extension of ADMM is not necessarily convergent. On the other hand, despite the lack of convergence, the direct extension of ADMM is empirically efficient for many applications. Thus we are interested in such an algorithm that can be implemented as easily as the direct extension of ADMM, while with comparable or even better numerical performance and guaranteed convergence. In this paper, we suggest correcting the output of the direct extension of ADMM slightly by a simple correction step. The correction step is simple in the sense that it is completely free from step-size computing and its step size is bounded away from zero for any iterate. A prototype algorithm in this prediction-correction framework is proposed; and a unified and easily checkable condition to ensure the convergence of this prototype algorithm is given. Theoretically, we show the contraction property, prove the global convergence and establish the worst-case convergence rate measured by the iteration complexity for this prototype algorithm. The analysis is conducted in the variational inequality context. Then, based on this prototype algorithm, we propose a class of specific ADMM-based algorithms that can be used for three-block separable convex minimization models. Their numerical efficiency is verified by an image decomposition problem.

37 citations


Journal ArticleDOI
TL;DR: Two separate analyses for the possible CG degeneracy are detailed, relying on a novel recently proposed theory, which includes an additional number, namely grossone, showing that grossone may represent a fruitful and promising tool to be exploited within Nonlinear Programming.
Abstract: This paper deals with an analysis of the Conjugate Gradient (CG) method (Hestenes and Stiefel in J Res Nat Bur Stand 49:409–436, 1952), in the presence of degenerates on indefinite linear systems. Several approaches have been proposed in the literature to issue the latter drawback in optimization frameworks, including reformulating the original linear system or recurring to approximately solving it. All the proposed alternatives seem to rely on algebraic considerations, and basically pursue the idea of improving numerical efficiency. In this regard, here we sketch two separate analyses for the possible CG degeneracy. First, we start detailing a more standard algebraic viewpoint of the problem, suggested by planar methods. Then, another algebraic perspective is detailed, relying on a novel recently proposed theory, which includes an additional number, namely grossone. The use of grossone allows to work numerically with infinities and infinitesimals. The results obtained using the two proposed approaches perfectly match, showing that grossone may represent a fruitful and promising tool to be exploited within Nonlinear Programming.

36 citations


Journal ArticleDOI
TL;DR: A Fokker–Planck control strategy for collective motion is investigated and the resulting optimality system is discretized using an alternate-direction implicit Chang–Cooper scheme that guarantees conservativeness, positivity, and second-order accuracy of the forward solution.
Abstract: A Fokker–Planck control strategy for collective motion is investigated. This strategy is formulated as the minimisation of an expectation objective with a bilinear optimal control problem governed by the Fokker–Planck equation modelling the evolution of the probability density function of the stochastic motion. Theoretical results on existence and regularity of optimal controls are provided. The resulting optimality system is discretized using an alternate-direction implicit Chang–Cooper scheme that guarantees conservativeness, positivity, $$L^1$$ stability, and second-order accuracy of the forward solution. A projected non-linear conjugate gradient scheme is used to solve the optimality system. Results of numerical experiments validate the theoretical accuracy estimates and demonstrate the efficiency of the proposed control framework.

Journal ArticleDOI
TL;DR: In this article, a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space is presented, which can be viewed as an adequate modification of the Douglas-Rachford method that yields a solution to the best approximation problem.
Abstract: In this paper we present a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method, termed AAMR for averaged alternating modified reflections, can be viewed as an adequate modification of the Douglas–Rachford method that yields a solution to the best approximation problem. Under a constraint qualification at the point of interest, we show strong convergence of the method. In fact, the so-called strong CHIP fully characterizes the convergence of the AAMR method for every point in the space. We report some promising numerical experiments where we compare the performance of AAMR against other projection methods for finding the closest point in the intersection of pairs of finite dimensional subspaces.

Journal ArticleDOI
TL;DR: The gradient projection method is revisited in the framework of nonlinear optimal control problems with bang–bang solutions and the strong convergence of the iterative sequence of controls and the corresponding trajectories is obtained.
Abstract: We revisit the gradient projection method in the framework of nonlinear optimal control problems with bang-bang solutions. We obtain the strong convergence of the iterative sequence of controls and the corresponding trajectories. Moreover, we establish a convergence rate, depending on a constant appearing in the corresponding switching function and prove that this convergence rate estimate is sharp. Some numerical illustrations are reported confirming the theoretical results.

Journal ArticleDOI
TL;DR: This work addresses the solution of constrained nonlinear systems by new linesearch quasi-Newton methods based on a proper use of the projection map onto the convex constraint set and on a derivative-free and nonmonotone linesearch strategy.
Abstract: We address the solution of constrained nonlinear systems by new linesearch quasi-Newton methods. These methods are based on a proper use of the projection map onto the convex constraint set and on a derivative-free and nonmonotone linesearch strategy. The convergence properties of the proposed methods are presented along with a worst-case iteration complexity bound. Several implementations of the proposed scheme are discussed and validated on bound-constrained problems including gas distribution network models. The results reported show that the new methods are very efficient and competitive with an existing affine-scaling procedure.

Journal ArticleDOI
TL;DR: A new SDP relaxation involving a number of positive semidefinite matrices of dimension O(n) produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy to tackle the assignment problem.
Abstract: The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension $$n^2$$ where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension $$\mathcal {O}(n)$$ no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Alternating Direction Method of Multipliers in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension $$\mathcal {O}(n)$$ . The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy to tackle the assignment problem.

Journal ArticleDOI
TL;DR: An hybrid algorithm is developed by coupling a multi-objective DIRECT-type algorithm with an efficient derivative-free local algorithm that performs well on a real ship hull optimization problem and shows viability of the proposed approach.
Abstract: The paper is concerned with black-box nonlinear constrained multi-objective optimization problems. Our interest is the definition of a multi-objective deterministic partition-based algorithm. The main target of the proposed algorithm is the solution of a real ship hull optimization problem. To this purpose and in pursuit of an efficient method, we develop an hybrid algorithm by coupling a multi-objective DIRECT-type algorithm with an efficient derivative-free local algorithm. The results obtained on a set of “hard” nonlinear constrained multi-objective test problems show viability of the proposed approach. Results on a hull-form optimization of a high-speed catamaran (sailing in head waves in the North Pacific Ocean) are also presented. In order to consider a real ocean environment, stochastic sea state and speed are taken into account. The problem is formulated as a multi-objective optimization aimed at (i) the reduction of the expected value of the mean total resistance in irregular head waves, at variable speed and (ii) the increase of the ship operability, with respect to a set of motion-related constraints. We show that the hybrid method performs well also on this industrial problem.

Journal ArticleDOI
TL;DR: Numerical results on a set of test problems show the efficiency of the implicit filtering algorithm when used to find a single Pareto solution of the problem and show through numerical experience that the proposed algorithm improves the performance of DMS alone whenUsed to reconstruct the entire Pare to front.
Abstract: This paper is concerned with the definition of new derivative-free methods for box constrained multiobjective optimization. The method that we propose is a non-trivial extension of the well-known implicit filtering algorithm to the multiobjective case. Global convergence results are stated under smooth assumptions on the objective functions. We also show how the proposed method can be used as a tool to enhance the performance of the Direct MultiSearch (DMS) algorithm. Numerical results on a set of test problems show the efficiency of the implicit filtering algorithm when used to find a single Pareto solution of the problem. Furthermore, we also show through numerical experience that the proposed algorithm improves the performance of DMS alone when used to reconstruct the entire Pareto front.

Journal ArticleDOI
TL;DR: In this paper, the authors propose Lasserre type semidefinite relaxation methods for computing real and subjective eigenvalues of nonsymmetric tensors, and show that for any tensor that has finitely many real $$\mathtt {Z}$$.............. -eigenvalues, each of them can be computed by solving a finite sequence of semi-definite relaxations.
Abstract: This paper discusses the computation of real $$\mathtt {Z}$$ -eigenvalues and $$\mathtt {H}$$ -eigenvalues of nonsymmetric tensors. A generic nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. The number of $$\mathtt {H}$$ -eigenvalues is finite for all tensors. We propose Lasserre type semidefinite relaxation methods for computing such eigenvalues. For every tensor that has finitely many real $$\mathtt {Z}$$ -eigenvalues, we can compute all of them; each of them can be computed by solving a finite sequence of semidefinite relaxations. For every tensor, we can compute all its real $$\mathtt {H}$$ -eigenvalues; each of them can be computed by solving a finite sequence of semidefinite relaxations.

Journal ArticleDOI
TL;DR: In this article, the authors consider general nonlinear programming problems with cardinality constraints and apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points.
Abstract: We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize a risk measure under a cardinality constraint on the portfolio. Various risk measures are considered, in particular Value-at-Risk and Conditional Value-at-Risk under normal distribution of returns and their robust counterparts under moment conditions. For these investment problems formulated as nonlinear programming problems with cardinality constraints we perform a numerical study on a large number of simulated instances taken from the literature and illuminate the computational performance of the Scholtes-type regularization method in comparison to other considered solution approaches: a mixed-integer solver, a direct continuous reformulation solver and the Kanzow–Schwartz regularization method, which has already been applied to Markowitz portfolio problems.

Journal ArticleDOI
TL;DR: The analysis and computational results show that acceleration may not bring any benefit in the quasi-Newton setting, and a practical variant of this method is investigated by establishing a simple stopping criterion for the subproblem optimization.
Abstract: A general, inexact, efficient proximal quasi-Newton algorithm for composite optimization problems has been proposed by Scheinberg and Tang (Math Program 160:495–529, 2016) and a sublinear global convergence rate has been established. In this paper, we analyze the global convergence rate of this method, in the both exact and inexact settings, in the case when the objective function is strongly convex. We also investigate a practical variant of this method by establishing a simple stopping criterion for the subproblem optimization. Furthermore, we consider an accelerated variant, based on FISTA of Beck and Teboulle (SIAM 2:183–202, 2009), to the proximal quasi-Newton algorithm. Jiang et al. (SIAM 22:1042–1064, 2012) considered a similar accelerated method, where the convergence rate analysis relies on very strong impractical assumptions on Hessian estimates. We present a modified analysis while relaxing these assumptions and perform a numerical comparison of the accelerated proximal quasi-Newton algorithm and the regular one. Our analysis and computational results show that acceleration may not bring any benefit in the quasi-Newton setting.

Journal ArticleDOI
TL;DR: This work proposes several outer approximations of the solution set of absolute value equations, and presents conditions for unsolvability and for existence of exponentially many solutions, too, and compares them with the known conditions.
Abstract: In the recent years, there has been an intensive research of absolute value equations $$Ax-b=B|x|$$ . Various methods were developed, but less attention has been paid to approximating or bounding the solutions. We start filling this gap by proposing several outer approximations of the solution set. We present conditions for unsolvability and for existence of exponentially many solutions, too, and compare them with the known conditions. Eventually, we carried out numerical experiments to compare the methods with respect to computational time and quality of estimation. This helps in identifying the cases, in which the bounds are tight enough to determine the signs of the solution, and therefore also the solution itself.

Journal ArticleDOI
TL;DR: In this article, a Lagrangian decomposition algorithm was proposed by Dan Bienstock and Mark Zuckerberg for solving the LP relaxation of a class of open pit mine project scheduling problems, including the well-known Resource Constrained Project Scheduling Problem (RCPSP).
Abstract: We study a Lagrangian decomposition algorithm recently proposed by Dan Bienstock and Mark Zuckerberg for solving the LP relaxation of a class of open pit mine project scheduling problems. In this study we show that the Bienstock---Zuckerberg (BZ) algorithm can be used to solve LP relaxations corresponding to a much broader class of scheduling problems, including the well-known Resource Constrained Project Scheduling Problem (RCPSP), and multi-modal variants of the RCPSP that consider batch processing of jobs. We present a new, intuitive proof of correctness for the BZ algorithm that works by casting the BZ algorithm as a column generation algorithm. This analysis allows us to draw parallels with the well-known Dantzig---Wolfe decomposition (DW) algorithm. We discuss practical computational techniques for speeding up the performance of the BZ and DW algorithms on project scheduling problems. Finally, we present computational experiments independently testing the effectiveness of the BZ and DW algorithms on different sets of publicly available test instances. Our computational experiments confirm that the BZ algorithm significantly outperforms the DW algorithm for the problems considered. Our computational experiments also show that the proposed speed-up techniques can have a significant impact on the solve time. We provide some insights on what might be explaining this significant difference in performance.

Journal ArticleDOI
TL;DR: In this article, the authors developed a more general continuous bilinear program which incorporates vertex weights, and which applies to the coarse graphs that are generated in a multilevel compression of the original vertex separator problem.
Abstract: The Vertex Separator Problem for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets that satisfy specified size constraints. The Vertex Separator Problem was formulated in the paper 10.1016/j.ejor.2014.05.042 as a continuous (non-concave/non-convex) bilinear quadratic program. In this paper, we develop a more general continuous bilinear program which incorporates vertex weights, and which applies to the coarse graphs that are generated in a multilevel compression of the original Vertex Separator Problem. We develop a method for improving upon a given vertex separator by applying a Mountain Climbing Algorithm to the bilinear program using an incidence vector for the separator as a starting guess. Sufficient conditions are developed under which the algorithm can improve upon the starting guess after at most two iterations. The refinement algorithm is augmented with a perturbation technique to enable escapes from local optima and is embedded in a multilevel framework for solving large scale instances of the problem. The multilevel algorithm is shown through computational experiments to perform particularly well on communication and collaboration networks.

Journal ArticleDOI
TL;DR: A parallel large neighborhood search framework for finding high quality primal solutions for general mixed-integer programs (MIPs) with the dual objective of reducing infeasibility and optimizing with respect to the original objective.
Abstract: We present a parallel large neighborhood search framework for finding high quality primal solutions for general mixed-integer programs (MIPs). The approach simultaneously solves a large number of sub-MIPs with the dual objective of reducing infeasibility and optimizing with respect to the original objective. Both goals are achieved by solving restricted versions of two auxiliary MIPs, where subsets of the variables are fixed. In contrast to prior approaches, ours does not require a feasible starting solution. We leverage parallelism to perform multiple searches simultaneously, with the objective of increasing the effectiveness of our heuristic. We computationally compare the proposed framework with a state-of-the-art MIP solver in terms of solution quality, scalability, reproducibility, and parallel efficiency. Results show the efficacy of our approach in finding high quality solutions quickly both as a standalone primal heuristic and when used in conjunction with an exact algorithm.

Journal ArticleDOI
TL;DR: A novel outer approximation approach to efficiently solve OCPs in the case of certain semilinear elliptic PDEs with static integer controls over arbitrary combinatorial structures, where the nonlinear part of the PDE to be non-decreasing and convex.
Abstract: Optimal control problems (OCPs) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach is to first discretize the problem, which results in huge and typically nonconvex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we propose a novel outer approximation approach to efficiently solve such OCPs in the case of certain semilinear elliptic PDEs with static integer controls over arbitrary combinatorial structures, where we assume the nonlinear part of the PDE to be non-decreasing and convex. The basic idea is to decompose the OCP into an integer linear programming (ILP) master problem and a subproblem for calculating linear cutting planes. These cutting planes rely on the pointwise concavity or submodularity of the PDE solution with respect to the control variables. The decomposition allows us to use standard solution techniques for ILPs as well as for PDEs. We further benefit from reoptimization strategies for the PDE solution due to the iterative structure of the algorithm. Experimental results show that the new approach is capable of solving the combinatorial OCP of a semilinear Poisson equation with up to 180 binary controls to global optimality within a 5 h time limit. In the case of the screened Poisson equation, which yields semi-infinite integer linear programs, problems with as many as 1400 binary controls are solved.

Journal ArticleDOI
TL;DR: Numerical experience on an image blind deconvolution problem show the improvements obtained by adopting a variable number of inner block iterations combined with a variable metric in the computation of the proximal operator.
Abstract: In this paper we propose an alternating block version of a variable metric linesearch proximal gradient method. This algorithm addresses problems where the objective function is the sum of a smooth term, whose variables may be coupled, plus a separable part given by the sum of two or more convex, possibly nonsmooth functions, each depending on a single block of variables. Our approach is characterized by the possibility of performing several proximal gradient steps for updating every block of variables and by the Armijo backtracking linesearch for adaptively computing the steplength parameter. Under the assumption that the objective function satisfies the Kurdyka-Łojasiewicz property at each point of its domain and the gradient of the smooth part is locally Lipschitz continuous, we prove the convergence of the iterates sequence generated by the method. Numerical experience on an image blind deconvolution problem show the improvements obtained by adopting a variable number of inner block iterations combined with a variable metric in the computation of the proximal operator.

Journal ArticleDOI
TL;DR: By discretizing the objective functional with the help of appropriate numerical quadrature formulas, it is able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution.
Abstract: We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included.

Journal ArticleDOI
TL;DR: This work investigates Euler discretization for a class of optimal control problems with a nonlinear cost functional of Mayer type, a non linear system equation with control appearing linearly and constraints defined by lower and upper bounds for the controls.
Abstract: We investigate Euler discretization for a class of optimal control problems with a nonlinear cost functional of Mayer type, a nonlinear system equation with control appearing linearly and constraints defined by lower and upper bounds for the controls. Under the assumption that the cost functional satisfies a growth condition we prove for the discrete solutions Holder type error estimates w.r.t. the mesh size of the discretization. If a stronger second-order optimality condition is satisfied the order of convergence can be improved. Numerical experiments confirm the theoretical findings.

Journal ArticleDOI
TL;DR: A novel randomized block coordinate descent method for the minimization of a convex composite objective function that uses (approximate) partial second-order (curvature) information, so that the algorithm performance is more robust when applied to highly nonseparable or ill conditioned problems.
Abstract: We present a novel randomized block coordinate descent method for the minimization of a convex composite objective function. The method uses (approximate) partial second-order (curvature) information, so that the algorithm performance is more robust when applied to highly nonseparable or ill conditioned problems. We call the method Flexible Coordinate Descent (FCD). At each iteration of FCD, a block of coordinates is sampled randomly, a quadratic model is formed about that block and the model is minimized approximately/inexactly to determine the search direction. An inexpensive line search is then employed to ensure a monotonic decrease in the objective function and acceptance of large step sizes. We present several high probability iteration complexity results to show that convergence of FCD is guaranteed theoretically. Finally, we present numerical results on large-scale problems to demonstrate the practical performance of the method.

Journal ArticleDOI
TL;DR: This work focuses on a new class of semidefinite relaxations, based on matrices of order 2n+1, which provide a good compromise between quality of the bound and computational effort to actually compute it, and exploits those bounds to obtain bounds on the size of the vertex separators.
Abstract: We consider graph three-partitions with the objective of minimizing the number of edges between the first two partition sets while keeping the size of the third block small. We review most of the existing relaxations for this min-cut problem and focus on a new class of semidefinite relaxations, based on matrices of order $$2n+1$$ which provide a good compromise between quality of the bound and computational effort to actually compute it. Here, n is the order of the graph. Our numerical results indicate that the new bounds are quite strong and can be computed for graphs of medium size ( $$n \approx 300$$ ) with reasonable effort of a few minutes of computation time. Further, we exploit those bounds to obtain bounds on the size of the vertex separators. A vertex separator is a subset of the vertex set of a graph whose removal splits the graph into two disconnected subsets. We also present an elegant way of convexifying non-convex quadratic problems by using semidefinite programming. This approach results with bounds that can be computed with any standard convex quadratic programming solver.