A globally convergent non-interior point algorithm with full Newton step for second-order cone programming

A non-interior point algorithm based on projection for second-order cone programming problems is proposed and analyzed. The main idea of the algorithm is that we cast the complementary equation in the primal-dual optimality conditions as a projection equation. By using this reformulation, we only need to solve a system of linear equations with the same coefficient matrix and compute two simple projections at each iteration, without performing any line search. This algorithm can start from an arbitrary point, and does not require the row vectors of A to be linearly independent. We prove that our algorithm is globally convergent under weak conditions. Preliminary numerical results demonstrate the effectiveness of our algorithm.     

Polynomiality of an inexact infeasible interior point algorithm for semidefinite programming

In this paper we present a primal-dual inexact infeasible interior-point algorithm for semidefinite programming problems (SDP). This algorithm allows the use of search directions that are calculated from the defining linear system with only moderate accuracy, and does not require feasibility to be maintained even if the initial iterate happened to be a feasible solution of the problem. Under a mild assumption on the inexactness, we show that the algorithm can find an -approximate solution of an SDP in O(n²ln(1/)) iterations. This bound of our algorithm is the same as that of the exact infeasible interior point algorithms proposed by Y. Zhang.Research supported in part by the Singapore-MIT alliance, and NUS Academic Research Grant R-146-000-032-112.Mathematics Subject Classification (1991): 90C05, 90C30, 65K05     

Logarithmic barrier decomposition-based interior point methods for stochastic symmetric programming

We introduce and study two-stage stochastic symmetric programs with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a Benders’ decomposition-based interior point algorithm for solving these problems and prove its polynomial complexity. Our convergence analysis proved by showing that the log barrier associated with the recourse function of stochastic symmetric programs behaves a strongly self-concordant barrier and forms a self-concordant family on the first stage solutions. Since our analysis applies to all symmetric cones, this algorithm extends Zhao’s results [G. Zhao, A log barrier method with Benders’ decomposition for solving two-stage stochastic linear programs, Math. Program. Ser. A 90 (2001) 507–536] for two-stage stochastic linear programs, and Mehrotra and Özevin’s results [S. Mehrotra, M.G. Özevin, Decomposition-based interior point methods for two-stage stochastic semidefinite programming, SIAM J. Optim. 18 (1) (2007) 206–222] for two-stage stochastic semidefinite programs.     

An interior point algorithm for semi-infinite linear programming

We consider the generalization of a variant of Karmarkar's algorithm to semi-infinite programming. The extension of interior point methods to infinite-dimensional linear programming is discussed and an algorithm is derived. An implementation of the algorithm for a class of semi-infinite linear programs is described and the results of a number of test problems are given. We pay particular attention to the problem of Chebyshev approximation. Some further results are given for an implementation of the algorithm applied to a discretization of the semi-infinite linear program, and a convergence proof is given in this case.     

Computational results of an interior point algorithm for large scale linear programming

This paper gives computational results for an efficient implementation of a variant of dual projective algorithm for linear programming. The implementation uses the preconditioned conjugate gradient method for computing projections. Our computational experience reported in this paper indicates that this algorithm has potential as an alternative for solving very large LPs in which the direct methods fail due to memory and CPU time requirements. The conjugate gradient algorithm was able to find very accurate directions even when the system was ill-conditioned. The paper also discusses a new mathematical technique called the reciprocal estimates for estimating the primal variables. We have conducted extensive computational experiments on problems representative of large classes of applications of current interest. We have also chosen instances of the problems of future potential interest, which could not be solved in the past due to the weakness of the prior solution methods, but which represent a large class of new applications. The hypergraph model is such an example. Comparison of our implementation with MINOS 5.1 shows that our implementation is orders of magnitude faster than MINOS 5.1 for these problems.     

Product-form Cholesky factorization in interior point methods for second-order cone programming

Second-order cone programming (SOCP) problems are typically solved by interior point methods. As in linear programming (LP), interior point methods can, in theory, solve SOCPs in polynomial time and can, in practice, exploit sparsity in the problem data. Specifically, when cones of large dimension are present, the density that results in the normal equations that are solved at each iteration can be remedied in a manner similar to the treatment of dense columns in an LP. Here we propose a product-form Cholesky factorization (PFCF) approach, and show that it is more numerically stable than the alternative Sherman-Morrison-Woodbury approach. We derive several PFCF variants and compare their theoretical perfomance. Finally, we prove that the elements of L in the Cholesky factorizations LDL^T that arise in interior point methods for SOCP are uniformly bounded as the duality gap tends to zero as long as the iterates remain is some conic neighborhood of the cental path.Mathematics Subject Classification (1991): 90C25, 90C51, 15A23Research supported in part by NSF Grants CDA 97-26385, DMS 01-04282, ONR Grant N000140310514 and DOE Grant GE-FG01-92ER-25126     

An infeasible full-NT step interior point algorithm for CQSCO

In this paper, based on the complex-symmetric and skew-Hermitian splitting (CSS) of the coefficient matrix, a modified complex-symmetric and skew-Hermitian-splitting (MCSS) iteration method is presented to solve a class of complex-symmetric indefinite linear systems from the classical state-space formulation of frequency analysis of the degree-of-freedom discrete system. The convergence properties of the MCSS method are obtained. The corresponding MCSS preconditioner is proposed and some useful properties of the preconditioned matrix are established. Numerical experiments are reported to verify the efficiency of both the MCSS method and the MCSS preconditioner.     

A class of nonlinear Lagrangians for nonconvex second order cone programming

This paper focuses on the study of a class of nonlinear Lagrangians for solving nonconvex second order cone programming problems. The nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence properties for the nonlinear Lagrange method are discussed when subproblems are assumed to be solved exactly and inexactly, respectively. The convergence theorems show that, under the second order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in nonlinear Lagrangian methods for nonlinear programming, we have to deal with the sigma term in the convergence analysis. Finally, we report numerical results by using modified Frisch’s function, modified Carroll’s function and the Log-Sigmoid function.     

A globally convergent interior point algorithm for non-convex nonlinear programming

In this paper, a new algorithm for tracing the combined homotopy path of the non-convex nonlinear programming problem is proposed. The algorithm is based on the techniques of β$\beta$-cone neighborhood and a combined homotopy interior point method. The residual control criteria, which ensures that the obtained iterative points are interior points, is given by the condition that ensures the β$\beta$-cone neighborhood to be included in the interior part of the feasible region. The global convergence and polynomial complexity are established under some hypotheses.     

A parallel interior point algorithm for linear programming on a network of transputers

Interior Point algorithms have become a very successful tool for solving large-scale linear programming problems. The Dual Affine algorithm is one of the Interior Point algorithms implemented in the computer program OB1. It is a good candidate for implementation on a parallel computer because it is very computing-intensive. A parallel Dual Affine algorithm is presented which is suitable for a parallel computer with a distributed memory. The algorithm obtains its speedup from parallel sparse linear algebra computations such as Cholesky factorisation, matrix multiplication, and triangular system solving, which form the bulk of the computing work. Efficient algorithms based on the grid distribution of matrices are presented for each of these computations. The algorithm is implemented in occam 2 on a square mesh of transputers. The resulting parallel program is connected to the sequentialFortran 77 program OB1, which performs the preprocessing and the postprocessing. Experimental results on a mesh of 400 transputers are given for a test set of seven realistic planning and scheduling problems from Shell and seven problems from the NETLIB LP collection; the results show a speedup of 88 for the largest problem.     

Second order symmetric duality in nondifferentiable multiobjective fractional programming with cone convex functions

In the present paper, we consider Mond-Weir type nondifferentiable second order fractional symmetric dual programs over arbitrary cones and derive duality results under second order K?F-convexity/K?F-pseudoconvexity assumptions. Our results generalize several known results in the literature.     

A trust region interior point algorithm for infinite dimensional nonlinear programming

A trust region interior point algorithm for infinite dimensional nonlinear problem, which is motivated by the application of black-box approach to the distributed parameter system optimal control problem with equality and inequality constraints on states and controls, and with bounds on the controls is formulated. By introducing a proper functional which is analogous to the Lagrange function, both equality and inequality constraints can be treated identically and the first order optimality condition is given, then based on the works of Coleman, Ulbrich and Heinkenschloss, the trust region interior point algorithm which is employed to solve the optimization problem under consideration is presented.     

二阶锥规划的基于自协调指数核函数的原始-对偶内点算法

基于一个自协调指数核函数, 设计求解二阶锥规划的原始-对偶内点算法. 根据自协调指数核函数的二阶导数与三阶导数的特殊关系, 在求解问题的中心路径时, 用牛顿方向代替了负梯度方向来确定搜索方向. 由于自协调指数核函数不具有``Eligible'性质, 在分析算法的迭代界时, 利用牛顿方法求解目标函数满足自协调性质的无约束优化问题的技术, 估计算法内迭代中自协调指数核函数确定的障碍函数的下降量, 得到原始-对偶内点算法大步校正的迭代界O(2N\frac{\log2N}{\varepsilon}), 这里N是二阶锥的个数. 这个迭代界与线性规划情形下的迭代界一致. 最后, 通过数值算例验证了算法的有效性.     

Non-differentiable symmetric duality for multiobjective programming with cone constraints

Two pairs of non-differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones, which are Wolfe type and Mond–Weir type, are considered. On the basis of weak efficiency with respect to a convex cone, we obtain symmetric duality results for the two pairs of problems under cone-invexity and cone-pseudoinvexity assumptions on the involved functions. Our results extend the results in Khurana [S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research 165 (2005) 592–597] to the non-differentiable multiobjective symmetric dual problem.     

Learning the optimal kernel for Fisher discriminant analysis via second order cone programming

Kernel Fisher discriminant analysis (KFDA) is a popular classification technique which requires the user to predefine an appropriate kernel. Since the performance of KFDA depends on the choice of the kernel, the problem of kernel selection becomes very important. In this paper we treat the kernel selection problem as an optimization problem over the convex set of finitely many basic kernels, and formulate it as a second order cone programming (SOCP) problem. This formulation seems to be promising because the resulting SOCP can be efficiently solved by employing interior point methods. The efficacy of the optimal kernel, selected from a given convex set of basic kernels, is demonstrated on UCI machine learning benchmark datasets.     

Second order duality for multiobjective programming with cone constraints

We focus on second order duality for a class of multiobjective programming problem subject to cone constraints. Four types of second order duality models are formulated. Weak and strong duality theorems are established in terms of the generalized convexity, respectively. Converse duality theorems, essential parts of duality theory, are presented under appropriate assumptions. Moreover, some deficiencies in the work of Ahmad and Agarwal (2010) are discussed.     

An active-set strategy in an interior point method for linear programming

We will present a potential reduction method for linear programming where only the constraints with relatively small dual slacks—termed active constraints—will be taken into account to form the ellipsoid constraint at each iteration of the process. The algorithm converges to the optimal feasible solution in O( L) iterations with the same polynomial bound as in the full constraints case, wheren is the number of variables andL is the data length. If a small portion of the constraints is active near the optimal solution, the computational cost to find the next direction of movement in one iteration may be considerably reduced by the proposed strategy.This research was partially done in June 1990 while the author was visiting the Department of Mathematics, University of Pisa.     

Proximal point algorithm with Schur decomposition on the cone of symmetric semidefinite positive matrices

In this work, we propose a proximal algorithm for unconstrained optimization on the cone of symmetric semidefinite positive matrices. It appears to be the first in the proximal class on the set of methods that convert a Symmetric Definite Positive Optimization in Nonlinear Optimization. It replaces the main iteration of the conceptual proximal point algorithm by a sequence of nonlinear programming problems on the cone of diagonal definite positive matrices that has the structure of the positive orthant of the Euclidian vector space. We are motivated by results of the classical proximal algorithm extended to Riemannian manifolds with nonpositive sectional curvature. An important example of such a manifold is the space of symmetric definite positive matrices, where the metrics is given by the Hessian of the standard barrier function −lndet(X). Observing the obvious fact that proximal algorithms do not depend on the geodesics, we apply those ideas to develop a proximal point algorithm for convex functions in this Riemannian metric.