This paper presents an exact penalty method for solving optimization problems with very general constraints covering, in particular, nonlinear programming (NLP), semidefinite programming (SDP), and second-ordercone p...
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This paper presents an exact penalty method for solving optimization problems with very general constraints covering, in particular, nonlinear programming (NLP), semidefinite programming (SDP), and second-order cone programming (SOCP). The algorithm is called the sequential linear cone method (SLCM) because for SDP and SOCP the main cost of computation amounts to solving at each iteration a linear cone program for which efficient solvers are available. Restricted to NLP, SLCM is exactly a sequential quadratic program method. Under two basic conditions which concern only the data, it is proved that the sequence of iterates is bounded. Furthermore, in particular, when the feasible set is nonempty, under two additional constraint qualification conditions, it is proved that the cluster points are stationary points. In that case, it is established also that the sequence of penalty parameters eventually stays constant, and for a particular class of data it is proved that a unit step length can be obtained.
second-ordercone programs are a class of convex optimization problems. We refer to them as deterministic second-ordercone programs (DSCOPs) since data defining them are deterministic. In DSOCPs we minimize a linear ...
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second-ordercone programs are a class of convex optimization problems. We refer to them as deterministic second-ordercone programs (DSCOPs) since data defining them are deterministic. In DSOCPs we minimize a linear objective function over the intersection of an affine set and a product of second-order (Lorentz) cones. Stochastic programs have been studied since 1950s as a tool for handling uncertainty in data defining classes of optimization problems such as linear and quadratic programs. Stochastic second-ordercone programs (SSOCPs) with recourse is a class of optimization problems that defined to handle uncertainty in data defining DSOCPs. In this paper we describe four application models leading to SSOCPs. (C) 2011 Elsevier Inc. All rights reserved.
In this paper, we present a new one-step smoothing Newton method for solving the second-order cone programming (SOCP). Based on a new smoothing function of the well-known Fischer-Burmeister function, the SOCP is appro...
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In this paper, we present a new one-step smoothing Newton method for solving the second-order cone programming (SOCP). Based on a new smoothing function of the well-known Fischer-Burmeister function, the SOCP is approximated by a family of parameterized smooth equations. Our algorithm solves only one system of linear equations and performs only one Armijo-type line search at each iteration. It can start from an arbitrary initial point and does not require the iterative points to be in the sets of strictly feasible solutions. Without requiring strict complementarity at the SOCP solution, the proposed algorithm is shown to be globally and locally quadratically convergent under suitable assumptions. Numerical experiments demonstrate the feasibility and efficiency of our algorithm.
This work addresses the issue of high dimensionality for linear multiclass Support Vector Machines (SVMs) using second-order cone programming (SOCP) formulations. These formulations provide a robust and efficient fram...
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This work addresses the issue of high dimensionality for linear multiclass Support Vector Machines (SVMs) using second-order cone programming (SOCP) formulations. These formulations provide a robust and efficient framework for classification, while an adequate feature selection process may improve predictive performance. We extend the ideas of SOCP-SVM from binary to multiclass classification, while a sequential backward elimination algorithm is proposed for variable selection, defining a contribution measure to determine the feature relevance. Experimental results with multiclass microarray datasets demonstrate the effectiveness of a low-dimensional data representation in terms of performance.
It is an extremely complex process of controlling walking motion for humanoid robot, and its dynamics model has many rich features. The article puts forward a kind of optimization design method on any time humanoid ro...
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It is an extremely complex process of controlling walking motion for humanoid robot, and its dynamics model has many rich features. The article puts forward a kind of optimization design method on any time humanoid robot walking movement. Firstly, this article makes a stability analysis of walking humanoid robot based on the ZMP criterion, at the same time with the design of using a humanoid robot walking movement by the criterion of the error between the expectations and kinetic energy of the weighted kinetic minimum norm make the problem into a second-order cone programming (SOCP) optimization problem, then use the interior point method to solve the kinetic energy of optimization coefficient of the humanoid robot walking movement, and by compares with the LMS design method and genetic algorithms, finally the algorithm is validated in the simulation and experiment, the numerical results are present for illustration.
A method based on mathematical programming is proposed for large deformation and contact analysis of cable networks. By explicitly considering these nonsmooth behaviors, we formulate the linear complementarity problem...
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A method based on mathematical programming is proposed for large deformation and contact analysis of cable networks. By explicitly considering these nonsmooth behaviors, we formulate the linear complementarity problems over symmetric cones under some practically acceptable assumptions. We also present the equivalent second-order cone programming ( SOCP) problems, which can be regarded as the minimization problem of total potential energy and complementary energy with the subsidiary constraints on the displacements and contact forces, respectively. By solving the presented SOCP problems by using the primal-dual interior-point method, the equilibrium configurations and internal forces of several cable networks are obtained without any assumptions on stress states and contact conditions.
A new numerical scheme for critical state elastoplasticity is presented and detailed with special reference to the modified Cam clay model. The scheme is based on an incremental variational formulation whose discrete ...
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A new numerical scheme for critical state elastoplasticity is presented and detailed with special reference to the modified Cam clay model. The scheme is based on an incremental variational formulation whose discrete approximation gives rise to a second-ordercone program that is solved using a newly developed algorithm. A number of examples demonstrating the capabilities of the new scheme are given. (C) 2011 Published by Elsevier B.V.
In Andreani et al. (Weak notions of nondegeneracy in nonlinear semidefinite programming, 2020), the classical notion of nondegeneracy (or transversality) and Robinson's constraint qualification have been revisited...
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In Andreani et al. (Weak notions of nondegeneracy in nonlinear semidefinite programming, 2020), the classical notion of nondegeneracy (or transversality) and Robinson's constraint qualification have been revisited in the context of nonlinear semidefinite programming exploiting the structure of the problem, namely its eigendecomposition. This allows formulating the conditions equivalently in terms of (positive) linear independence of significantly smaller sets of vectors. In this paper, we extend these ideas to the context of nonlinear second-order cone programming. For instance, for an m-dimensional second-ordercone, instead of stating nondegeneracy at the vertex as the linear independence of m derivative vectors, we do it in terms of several statements of linear independence of 2 derivative vectors. This allows embedding the structure of the second-ordercone into the formulation of nondegeneracy and, by extension, Robinson's constraint qualification as well. This point of view is shown to be crucial in defining significantly weaker constraint qualifications such as the constant rank constraint qualification and the constant positive linear dependence condition. Also, these conditions are shown to be sufficient for guaranteeing global convergence of several algorithms, while still implying metric subregularity and without requiring boundedness of the set of Lagrange multipliers.
Based on the Chen-Harker-Kanzow-Smale smoothing function, a non-interior continuation method is proposed for solving the second-order cone programming. Unlike interior point methods, the proposed algorithm can start f...
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Based on the Chen-Harker-Kanzow-Smale smoothing function, a non-interior continuation method is proposed for solving the second-order cone programming. Unlike interior point methods, the proposed algorithm can start from an arbitrary point. Our algorithm solves only one linear system of equations and performs only one line search at each iteration. Without uniform nonsingularity, the algorithm is shown to be globally and locally quadratically convergent. Numerical results indicate that our algorithm is promising.
The minimum principle of complementary energy is established for cable networks involving only stress components as variables in geometrically nonlinear elasticity. It is rather amazing that the complementary energy a...
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The minimum principle of complementary energy is established for cable networks involving only stress components as variables in geometrically nonlinear elasticity. It is rather amazing that the complementary energy always attains minimum value at the equilibrium state irrespective of the stability of cable networks, contrary to the fact that only the stationary principles have been presented for elastic trusses and continua even in the case of stable equilibrium state. In order to show the strong duality between the minimization problems of total potential energy and complementary energy, the convex formulations of these problems are investigated, which can be embedded into a primal-dual pair of second-order cone programming problems. The existence and uniqueness of solution are also investigated for the minimization problem of complementary energy. (C) 2003 Elsevier Ltd. All rights reserved.
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