In this paper, we show that the popular K-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but amenable to a tractabl...
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In this paper, we show that the popular K-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but amenable to a tractable semidefinite programming (SDP) relaxation that is tighter than the current SDP relaxation schemes in the literature. In contrast to the existing schemes, our proposed SDP formulation gives rise to solutions that can be leveraged to identify the clusters. We devise a new approximation algorithm for K-means clustering that utilizes the improved formulation and empirically illustrate its superiority over the state-of-the-art solution schemes.
The present paper is concerned with the simulation of the Casagrande test carried out on a rammed earth material for wall-type structures in the framework of Limit Analysis (LA). In a preliminary study, the material i...
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The present paper is concerned with the simulation of the Casagrande test carried out on a rammed earth material for wall-type structures in the framework of Limit Analysis (LA). In a preliminary study, the material is considered as a homogeneous Coulomb material, and existing LA static and kinematic codes are used for the simulation of the test. In each loading case, static and kinematic bounds coincide;the corresponding exact solution is a two-rigid-block mechanism together with a quasi-constant stress vector and a velocity jump also constant along the interface, for the three loading cases. In a second study, to take into account the influence of compressive loadings related to the porosity of the material, an elliptic criterion (denoted Cohesive Cam-Clay, CCC) is defined based on recent homogenization results about the hollow sphere model for porous Coulomb materials. Finally, original finite element formulations of the static and mixed kinematic methods for the CCC material are developed and applied to the Casagrande test. The results are the same than above, except that this time the velocity jump depends on the compressive loading, which is more realistic but not satisfying fully the experimental observations. Therefore, the possible extensions of this work towards non-standard direct methods are analyzed in the conclusion section. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS.
We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concor...
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We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concordant families with respect to barrier parameters. These results are used to develop primal decomposition-based interior-point algorithms. The worst case iteration complexity of the developed algorithms is shown to be the same as that for the short- and long-step primal interior algorithms applied to the extensive formulation of our problem.
This paper presents a novel equilibrium formulation, that uses the cell-based smoothed method and conic programming, for limit and shakedown analysis of structures. The virtual strains are computed using straining cel...
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This paper presents a novel equilibrium formulation, that uses the cell-based smoothed method and conic programming, for limit and shakedown analysis of structures. The virtual strains are computed using straining cell-based smoothing technique based on elements of discretizexl mesh. Fictitious elastic stresses are also determined within the framework of finite element method (CS-FEM)-based Galerkin procedure, and equilibrium equations for residual stresses are satisfied in an average sense at every cell-based smoothing cell. All constrains are imposed at only one point in the smoothing domains, instead of Gauss points as in a standard FEM-based procedure. The resulting optimization problem is then handled using the highly efficient solvers. Various numerical examples are investigated, and obtained solutions are compared with available results in the literature.
We consider mathematical programs with complementarity constraints in Banach spaces. In particular, we focus on the situation that the complementarity constraint is defined by a non-polyhedric cone K. We demonstrate h...
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We consider mathematical programs with complementarity constraints in Banach spaces. In particular, we focus on the situation that the complementarity constraint is defined by a non-polyhedric cone K. We demonstrate how strong stationarity conditions can be obtained in an abstract setting. These conditions and their verification can be made more precise in the case that Z is a Hilbert space and if the projection onto K is directionally differentiable with a derivative as given in Haraux (Journal of the Mathematical Society of Japan 29(4), 615-631, 1977, Theorem 1). Finally, we apply the theory to optimization problems with semidefinite and second-order-cone complementarity constraints. We obtain that local minimizers are strongly stationary under a variant of the linear-independence constraint qualification, and these are novel results.
We study solution approaches to a class of mixed-integer nonlinear programming problems that arise from recent developments in risk-averse stochastic optimization and contain second- and p-order cone programming as sp...
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We study solution approaches to a class of mixed-integer nonlinear programming problems that arise from recent developments in risk-averse stochastic optimization and contain second- and p-order cone programming as special cases. We explore possible applications of some of the solution techniques that have been successfully used in mixed-integer conic programming and show how they can be generalized to the problems under consideration. Particularly, we consider a branch-and-bound method based on outer polyhedral approximations, lifted nonlinear cuts, and linear disjunctive cuts. Results of numerical experiments with discrete portfolio optimization models are presented. (C) 2016 Elsevier B.V. All rights reserved.
In this paper a special semi-smooth equation associated to the second order cone is studied. It is shown that, under mild assumptions, the semi-smooth Newton method applied to this equation is well-defined and the gen...
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In this paper a special semi-smooth equation associated to the second order cone is studied. It is shown that, under mild assumptions, the semi-smooth Newton method applied to this equation is well-defined and the generated sequence is globally and Q-linearly convergent to a solution. As an application, the obtained results are used to study the linear second order cone complementarity problem, with special emphasis on the particular case of positive definite matrices. Moreover, some computational experiments designed to investigate the practical viability of the method are presented. (C) 2016 Published by Elsevier Inc.
To deal with uncertainties of renewable energy,demand and price signals in real-time microgrid operation,this paper proposes a model predictive control strategy for microgrid economic dispatch, where hourly schedule i...
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To deal with uncertainties of renewable energy,demand and price signals in real-time microgrid operation,this paper proposes a model predictive control strategy for microgrid economic dispatch, where hourly schedule is constantly optimized according to the current system state and latest forecast information. Moreover, implicit network topology of the microgrid and corresponding power flow constraints are considered, which leads to a mixed integer nonlinear optimal power flow problem. Given the non-convexity feature of the original problem, the technique of conic programming is applied to efficiently crack the nut. Simulation results from a reconstructed IEEE-33 bus system and comparisons with the routine day-ahead microgrid schedule sufficiently substantiate the effectiveness of the proposed MPC strategy and the conic programming method.
Let K be a closed convex cone with dual We say that a linear transformation The dimension of the space of all such transformations is called the Lyapunov rank ofK. This number was introduced and studied by Rudolf etal...
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Let K be a closed convex cone with dual We say that a linear transformation The dimension of the space of all such transformations is called the Lyapunov rank ofK. This number was introduced and studied by Rudolf etal.[Bilinear optimality constraints for the cone of positive polynomials, Math. Program., Ser. B 129 (2011), pp. 5-31] for proper cones because of its connection to conic programming and complementarity problems. The assumption that K is proper turns out to be nonessential. We first develop the basic theory for cones that are merely closed and convex. We then devise a way to compute the Lyapunov rank of any closed convex cone and show that the Lyapunov-like transformations on a closed convex cone are related to the Lie algebra of its automorphism group. Next, we extend some results for proper polyhedral cones. Finally, we devise algorithms to compute both the space of all Lyapunov-like transformations and the Lyapunov rank of a polyhedral closed convex cone.
In this paper the hollow sphere model is investigated within the framework of limit analysis (LA), using the classical two-part velocity field, i.e., the exact solution for hydrostatic loading plus a linear solution, ...
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In this paper the hollow sphere model is investigated within the framework of limit analysis (LA), using the classical two-part velocity field, i.e., the exact solution for hydrostatic loading plus a linear solution, in both cases of von Mises and Drucker-Prager matrices. We use the kinematic LA approach in a quasi-analytical approach by imposing the plastic admissibility (PA) condition (Drucker-Prager matrix) or upper bounding the dissipated power (von Mises matrix) in a sufficiently high number of distributed points, thanks to conic programming formulations. Then we analyze the "porous Drucker-Prager" case to confirm that the so-called UBM (Upper Bound Model) approach of Guo et al. (2008) is only an estimate, although a good one in fact. Moreover, it is shown that the mean stress axis should not be a strict axis of symmetry for the macroscopic criterion. Then, considering the "Porous von Mises" case, we obtain that the real criterion is not only lower than the Gurson criterion, but probably non-symmetric with respect to the mean stress axis, more than in the Drucker-Prager case. We finally use ad hoc updated 3D-FEM LA codes to confirm the previous results and to evaluate the entire influence of the third stress invariant: the classical (Sigma(m), Sigma(eqv)) formulation of the Gurson criterion clearly overestimates the real, non-symmetric solution of the hollow sphere model, at least for porosities of the same order of magnitude as the value used in this work. (C) 2010 Elsevier Masson SAS. All rights reserved.
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