Sanjay Mehrotra

On the Implementation of a Primal-Dual Interior Point Method

This paper gives an approach to implementing a second-order primal-dual interior point method. It uses a Taylor polynomial of second order to approximate a primal-dual trajectory. The computations for the second derivative are combined with the computations for the centering direction. Computations in this approach do not require that primal and dual solutions be feasible. Expressions are given to compute all the higher-order derivatives of the trajectory of interest. The implementation ensures that a suitable potential function is reduced by a constant amount at each iteration.There are several salient features of this approach. An adaptive heuristic for estimating the centering parameter is given. The approach used to compute the step length is also adaptive. A new practical approach to compute the starting point is given. This approach treats primal and dual problems symmetrically.Computational results on a subset of problems available from netlib are given. On mutually tested problems the results show...

A branch-and-cut method for 0-1 mixed convex programming

Abstract.We generalize the disjunctive approach of Balas, Ceria, and Cornuéjols [2] and devevlop a branch-and-cut method for solving 0-1 convex programming problems. We show that cuts can be generated by solving a single convex program. We show how to construct regions similar to those of Sherali and Adams [20] and Lovász and Schrijver [12] for the convex case. Finally, we give some preliminary computational results for our method.

Solving symmetric indefinite systems in an interior-point method for linear programming

We describe an implementation of a primal—dual path following method for linear programming that solves symmetric indefinite “augmented” systems directly by Bunch—Parlett factorization, rather than reducing these systems to the positive definite “normal equations” that are solved by Cholesky factorization in many existing implementations. The augmented system approach is seen to avoid difficulties of numerical instability and inefficiency associated with free variables and with dense columns in the normal equations approach. Solving the indefinite systems does incur an extra overhead, whose median is about 40% in our tests; but the augmented system approach proves to be faster for a minority of cases in which the normal equations have relatively dense Cholesky factors. A detailed analysis shows that the augmented system factorization is reliable over a fairly large range of the parameter settings that control the tradeoff between sparsity and numerical stability.

PCx: an interior-point code for linear programming

We describe the code PCx, a primal-dual interior-point code for linear programming. Information is given about problem formulation and the underlying algorithm, along with instructions for installing, invoking, and using the code. Computational results on standard test problems are reported.

Finding an interior point in the optimal face of linear programs

We study the problem of finding a point in the relative interior of the optimal face of a linear program. We prove that in the worst case such a point can be obtained in O(n3L) arithmetic operations. This complexity is the same as the complexity for solving a linear program. We also show how to find such a point in practice. We report and discuss computational results obtained for the linear programming problems in the NETLIB test set.

On finding a vertex solution using interior point methods

Abstract An approach is proposed to generate a vertex solution while using a primal-dual interior point method to solve linear programs. A controlled random perturbation is made to the cost vector. A method to identify the active constraints at the vertex to which the solutions are converging is given. This basic method is further refined to save computational effort. The proposed approach is tested by using a variation of the primal-dual interior point method. Our method is developed by taking a predictor-corrector approach. In practice this method takes considerably fewer iterations to solve linear programs than methods described by Choi, Monma, and Shanno; Lustig, Marsten, and Shanno; and Domich, Boggs, Donaldson, and Witzgall. Computational results on problems from the NETLIB test set are reported to test our approach for finding vertex solutions. These results show that one perturbation is enough to force the solutions to converge to a vertex. The results indicate that the proposed approach is insensitive to the number of degenerate variables. The results also indicate that the effort required to generate a vertex solution is comparable to that required to solve the problem using an interior point method.

PCx user guide

We describe the code PCx, a primal-dual interior-point code for linear programming. Information is given about problem formulation and the underlying algorithm, along with instructions for installing, invoking, and using the code. Computational results on standard test problems are tabulated. The current version number is 1.0.

Sample average approximation of stochastic dominance constrained programs

In this paper we study optimization problems with second-order stochastic dominance constraints. This class of problems allows for the modeling of optimization problems where a risk-averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks. Here we deal with the case of multi-variate stochastic dominance under general distributions and nonlinear functions. We introduce the concept of

An algorithm for convex quadratic programming that requires O ( n 3.5 L ) arithmetic operations

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