共查询到20条相似文献,搜索用时 15 毫秒
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A class of iterative methods??filling methods??for polyhedral approximation of convex compact bodies is introduced and studied. In contrast to augmentation methods, the vertices of the approximating polytope can lie not only on the boundary of the body but also inside it. Within the proposed class, Hausdorff or H-methods of filling are singled out, for which the convergence rates (asymptotic and at the initial stage of the approximation) are estimated. For the approximation of nonsmooth convex compact bodies, the resulting convergence rate estimates coincide with those for augmentation H-methods. 相似文献
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Wies?awa T. Obuchowska 《European Journal of Operational Research》2012,218(1):58-67
In this paper we address the problem of the infeasibility of systems defined by reverse convex inequality constraints, where some or all of the variables are integer. In particular, we provide a polynomial algorithm that identifies a set of all constraints critical to feasibility (CF), that is constraints that may affect a feasibility status of the system after some perturbation of the right-hand sides. Furthermore, we will investigate properties of the irreducible infeasible sets and infeasibility sets, showing in particular that every irreducible infeasible set as well as infeasibility sets in the considered system, are subsets of the set CF of constraints critical to feasibility. 相似文献
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Henrik Schulz 《Discrete Applied Mathematics》2009,157(16):3485-3493
In this paper we introduce an algorithm for the creation of polyhedral approximations for certain kinds of digital objects in a three-dimensional space. The objects are sets of voxels represented as strongly connected subsets of an abstract cell complex. The proposed algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background voxels. The result of this method is a polyhedron which separates object voxels from background voxels. The objects processed by this algorithm and also the background voxel components inside the convex hull of the objects are restricted to have genus 0. The second aim of this paper is to present some practical improvements to the discussed convex hull algorithm to reduce computation time. 相似文献
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In this paper we consider the consistent partition problem in reverse convex and convex mixed-integer programming. In particular we will show that for the considered classes of convex functions, both integer and relaxed systems can be partitioned into two disjoint subsystems, each of which is consistent and defines an unbounded region. The polynomial time algorithm to generate the partition will be proposed and the algorithm for a maximal partition will also be provided. 相似文献
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Maxim V. Balashov 《Journal of Mathematical Analysis and Applications》2011,374(2):529-537
We consider polyhedral approximations of strictly convex compacta in finite-dimensional Euclidean spaces (such compacta are also uniformly convex). We obtain the best possible estimates for errors of considered approximations in the Hausdorff metric. We also obtain new estimates of an approximate algorithm for finding the convex hulls. 相似文献
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Two approximation algorithms for solving convex vector optimization problems (CVOPs) are provided. Both algorithms solve the CVOP and its geometric dual problem simultaneously. The first algorithm is an extension of Benson’s outer approximation algorithm, and the second one is a dual variant of it. Both algorithms provide an inner as well as an outer approximation of the (upper and lower) images. Only one scalar convex program has to be solved in each iteration. We allow objective and constraint functions that are not necessarily differentiable, allow solid pointed polyhedral ordering cones, and relate the approximations to an appropriate \(\epsilon \) -solution concept. Numerical examples are provided. 相似文献
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A coloring of the vertices of a graph G is convex if, for each assigned color d, the vertices with color d induce a connected subgraph of G. We address the convex recoloring problem, defined as follows. Given a graph G and a coloring of its vertices, recolor a minimum number of vertices of G, so that the resulting coloring is convex. This problem is known to be NP-hard even when G is a path. We show an integer programming formulation for the weighted version of this problem on arbitrary graphs, and then specialize it for trees. We study the facial structure of the polytope defined as the convex hull of the integer points satisfying the restrictions of the proposed ILP formulation, present several classes of facet-defining inequalities and discuss separation algorithms. 相似文献
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This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some other well-studied convex sets. Several classes of valid and facet-inducing inequalities are also derived. 相似文献
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Ivan Singer 《Journal of Mathematical Analysis and Applications》1979,69(2):571-584
We show that for a continuous convex functional f on a locally convex space E and a convex subset G of E such that inf f(E) < inf f(G), the problems of computing inf f(G) and of characterizing the elements g0?G with f(g0) = inf f(G) can be reduced to the same problems for a suitable hyperplane H0 in E. 相似文献
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Alberto Del Pia 《Mathematical Programming》2018,172(1-2):3-16
Concave mixed-integer quadratic programming is the problem of minimizing a concave quadratic polynomial over the mixed-integer points in a polyhedral region. In this work we describe an algorithm that finds an \(\epsilon \)-approximate solution to a concave mixed-integer quadratic programming problem. The running time of the proposed algorithm is polynomial in the size of the problem and in \(1/\epsilon \), provided that the number of integer variables and the number of negative eigenvalues of the objective function are fixed. The running time of the proposed algorithm is expected unless \(\mathcal {P}=\mathcal {NP}\). 相似文献
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In this paper we investigate the effects of replacing the objective function of a 0-1 mixed-integer convex program (MIP) with a “proximity” one, with the aim of using a black-box solver as a refinement heuristic. Our starting observation is that enumerative MIP methods naturally tend to explore a neighborhood around the solution of a relaxation. A better heuristic performance can however be expected by searching a neighborhood of an integer solution—a result that we obtain by just modifying the objective function of the problem at hand. The relationship of this approach with primal integer methods is also addressed. Promising computational results on different proof-of-concept implementations are presented, suggesting that proximity search can be quite effective in quickly refining a given feasible solution. This is particularly true when a sequence of similar MIPs has to be solved as, e.g., in a column-generation setting. 相似文献
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In many problems of linear image restoration, the point spread function is assumed to be known even if this information is usually not available. In practice, both the blur matrix and the restored image should be performed from the observed noisy and blurred image. In this case, one talks about the blind image restoration. In this paper, we propose a method for blind image restoration by using convex constrained optimization techniques for solving large-scale ill-conditioned generalized Sylvester equations. The blur matrix is approximated by a Kronecker product of two matrices having Toeplitz and Hankel forms. The Kronecker product approximation is obtained from an estimation of the point spread function. Numerical examples are given to show the efficiency of our proposed method. 相似文献
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A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm. 相似文献
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We present two linearization-based algorithms for mixed-integer nonlinear programs (MINLPs) having a convex continuous relaxation. The key feature of these algorithms is that, in contrast to most existing linearization-based algorithms for convex MINLPs, they do not require the continuous relaxation to be defined by convex nonlinear functions. For example, these algorithms can solve to global optimality MINLPs with constraints defined by quasiconvex functions. The first algorithm is a slightly modified version of the LP/NLP-based branch-and-bouund \((\text{ LP/NLP-BB })\) algorithm of Quesada and Grossmann, and is closely related to an algorithm recently proposed by Bonami et al. (Math Program 119:331–352, 2009). The second algorithm is a hybrid between this algorithm and nonlinear programming based branch-and-bound. Computational experiments indicate that the modified LP/NLP-BB method has comparable performance to LP/NLP-BB on instances defined by convex functions. Thus, this algorithm has the potential to solve a wider class of MINLP instances without sacrificing performance. 相似文献
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A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or quadratic integer-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original MINLP problem. After an initial overestimated set has been obtained the algorithm solves a sequence of mixed-integer linear programming or mixed-integer quadratic programming subproblems and refines the overestimated set by generating more supporting hyperplanes in each iteration. Compared to the extended cutting plane algorithm ESH generates a tighter overestimated set and unlike outer approximation the generation point for the supporting hyperplanes is found by a simple line search procedure. In this paper it is proven that the ESH algorithm converges to a global optimum for convex MINLP problems. The ESH algorithm is implemented as the supporting hyperplane optimization toolkit (SHOT) solver, and an extensive numerical comparison of its performance against other state-of-the-art MINLP solvers is presented. 相似文献
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Monika Ludwig 《Archiv der Mathematik》1994,63(4):377-384