We deal with the nonconvex program called Generalized Lattice Point Problem. Here, a linear function is to be minimized over such points of a polyhedron which belong to the at most q-dimensional faces of another polyh...
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We deal with the nonconvex program called Generalized Lattice Point Problem. Here, a linear function is to be minimized over such points of a polyhedron which belong to the at most q-dimensional faces of another polyhedron. We present a finite cutting plane algorithm for solving the considered problem. Computational experience is also provided.
We investigate a variational approach to nonpotential perturbations of gradient flows of nonconvex energies in Hilbert spaces. We prove existence of solutions to elliptic-in-time regularizations of gradient flows by c...
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We investigate a variational approach to nonpotential perturbations of gradient flows of nonconvex energies in Hilbert spaces. We prove existence of solutions to elliptic-in-time regularizations of gradient flows by combining the minimization of a parameter-dependent functional over entire trajectories and a fixed-point argument. These regularized solutions converge up to subsequences to solutions of the gradient flow as the regularization parameter goes to zero. Applications of the abstract theory to nonlinear reaction diffusion systems are presented. (C) 2016 Elsevier Inc. All rights reserved.
In this paper, an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming ( SDP) problems is described and analyzed. Such nonlinear and nonconvex programs arise, e. g., ...
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In this paper, an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming ( SDP) problems is described and analyzed. Such nonlinear and nonconvex programs arise, e. g., in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional Hilbert space. The algorithm treats the abstract states and controls as independent variables. In particular, an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear SDP-condition, a positive definiteness condition, and a nonlinear matrix equation is considered. The algorithm is designed to take advantage of the structure of the problem. It is an extension of an interior point trust region method to nonlinear and nonconvex SDPs, with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration. Some convergence results are given, and, finally, several numerical examples demonstrate the applicability of the considered algorithm.
Let f: X --> Y be a nonlinear differentiable map, X, Y are Hilbert spaces, B(a, r) is a ball in X with a center a and radius r, Suppose f'(x) is Lipschitz in B(a, r) with Lipschitz constant L and f'(a) is a...
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Let f: X --> Y be a nonlinear differentiable map, X, Y are Hilbert spaces, B(a, r) is a ball in X with a center a and radius r, Suppose f'(x) is Lipschitz in B(a, r) with Lipschitz constant L and f'(a) is a surjection: f(a)X = Y this implies the existence of v > 0 such that parallel tof'(a)*y parallel to greater than or equal to nu parallel toy parallel to, For Ally is an element of Y. Then, if epsilon < min[r, /(2L)}, the image F = f(B(a, epsilon)) of the ball B(a, epsilon) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint parallel tox - a parallel to less than or equal to epsilon. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for 'small power control' is convex. This leads to various results in optimal control.
The aim of this paper is to propose a solution method for the minimization of a class of generalized linear functions on a flow polytope. The problems will be solved by means of a network algorithm, based on graph ope...
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The aim of this paper is to propose a solution method for the minimization of a class of generalized linear functions on a flow polytope. The problems will be solved by means of a network algorithm, based on graph operations, which lies within the class of the so-called 'optimal level solutions' parametric methods. The use of the network structure of flow polytopes, allows to obtain good algorithm performances and small numerical errors. Results of a computational test are also provided.
Conditions alpha and beta are two well-known rationality conditions in the theory of rational choice. This paper examines the implications of weaker versions of these two rationality conditions in the context of solut...
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Conditions alpha and beta are two well-known rationality conditions in the theory of rational choice. This paper examines the implications of weaker versions of these two rationality conditions in the context of solutions to nonconvex bargaining problems. It is shown that, together with the standard axioms of efficiency and strict individual rationality, they imply rationalizability of solutions to nonconvex bargaining problems. We then characterize asymmetric Nash solutions by imposing a continuity and the scale invariance requirements. These results make a further connection between solutions to nonconvex bargaining problems and rationalizability of choice function in the theory of rational choice. (c) 2013 Elsevier B.V. All rights reserved.
This technical comment refers to the discussion of strong consistency of several bounding procedures in Lemma 2.1 and Proposition 2.1 of Ref. 1. A necessary clarification is given of the notion of convergence φq → ...
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This technical comment refers to the discussion of strong consistency of several bounding procedures in Lemma 2.1 and Proposition 2.1 of Ref. 1. A necessary clarification is given of the notion of convergence φq → φ in Lemma 2.1, and a derivation of Proposition 2.1 is presented that includes a new and simple consistency proof of the classical bounding by convex envelopes used in many branch-and-bound procedures.
In this note, a simple proof of a theorem concerning functions whose local minima are global is presented and some closedness properties of this class of functions are discussed.
In this note, a simple proof of a theorem concerning functions whose local minima are global is presented and some closedness properties of this class of functions are discussed.
Automatic discovery of community structures in complex networks is a fundamental task in many disciplines, including physics, biology, and the social sciences. The most used criterion for characterizing the existence ...
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Automatic discovery of community structures in complex networks is a fundamental task in many disciplines, including physics, biology, and the social sciences. The most used criterion for characterizing the existence of a community structure in a network is modularity, a quantitative measure proposed by Newman and Girvan (2004). The discovery community can be formulated as the so-called modularity maximization problem that consists of finding a partition of nodes of a network with the highest modularity. In this letter, we propose a fast and scalable algorithm called DCAM, based on DC (difference of convex function) programming and DCA (DC algorithms), an innovative approach in nonconvex programming framework for solving the modularity maximization problem. The special structure of the problem considered here has been well exploited to get an inexpensive DCA scheme that requires only a matrix-vector product at each iteration. Starting with a very large number of communities, DCAM furnishes, as output results, an optimal partition together with the optimal number of communities c*, that is, the number of communities is discovered automatically during DCAM's iterations. Numerical experiments are performed on a variety of real-world network data sets with up to 4,194,304 nodes and 30,359,198 edges. The comparative results with height reference algorithms show that the proposed approach outperforms them not only on quality and rapidity but also on scalability. Moreover, it realizes a very good trade-off between the quality of solutions and the run time.
Schedule optimization is crucial to reduce energy consumption of flexible manufacturing systems (FMSs) with shared resources and route flexibility. Based on the weighted p-timed Petri Net (WTPN) models of FMS, this pa...
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Schedule optimization is crucial to reduce energy consumption of flexible manufacturing systems (FMSs) with shared resources and route flexibility. Based on the weighted p-timed Petri Net (WTPN) models of FMS, this paper considers a scheduling problem which minimizes both productive and idle energy consumption subjected to general production constraints. The considered problem is proven to be a nonconvex mixed integer nonlinear program (MINLP). A new reachability graph (RG)-based discrete dynamic programming (DP) approach is proposed for generating near energy-optimal schedules within adequate computational time. The nonconvex MINLP is sampled, and the reduced RG is constructed such that only reachable paths are retained for computation of the energy-optimal path. Each scheduling subproblem is linearized, and each optimal substructure is computed to store in a routing table. It is proven that the sampling-induced error is bounded, and this upper bound can be reduced by increasing the sampling frequency. Experiment results on an industrial stamping system show the effectiveness of our proposed scheduling method in terms of computational complexity and deviation from optimality.
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