Gongyun Zhao

A Log-Barrier method with Benders decomposition for solving two-stage stochastic linear programs

Abstract.An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving two-stage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite number of scenarios, the algorithm is shown to converge globally and to run in polynomial-time.

On the complexity of following the central path of linear programs by linear extrapolation II

A class of algorithms is proposed for solving linear programming problems (withm inequality constraints) by following the central path using linear extrapolation with a special adaptive choice of steplengths. The latter is based on explicit results concerning the convergence behaviour of Newtons method to compute points on the central pathx(r), r>0, and this allows to estimate the complexity, i.e. the total numberN = N(R, δ) of steps needed to go from an initial pointx(R) to a final pointx(δ), R>δ>0, by an integral of the local “weighted curvature” of the (primal—dual) path. Here, the central curve is parametrized with the logarithmic penalty parameterr↓0. It is shown that for large classes of problems the complexity integral, i.e. the number of stepsN, is not greater than constmα log(R/δ), whereα < 1/2 e.g.α = 1/4 orα = 3/8 (note thatα = 1/2 gives the complexity of zero order methods). We also provide a lower bound for the complexity showing that for some problems the above estimation can hold only forα ⩾ 1/3.As a byproduct, many analytical and structural properties of the primal—dual central path are obtained: there are, for instance, close relations between the weighted curvature and the logarithmic derivatives of the slack variables; the dependence of these quantities on the parameterr is described. Also, related results hold for a family of weighted trajectories, into which the central path can be embedded.

Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization

We show that a locally Lipschitz homeomorphism function is semismooth at a given point if and only if its inverse function is semismooth at its image point. We present a sufficient condition for the semismoothness of solutions to generalized equations over cone reducible (nonpolyhedral) convex sets. We prove that the semismoothness of solutions to the Moreau-Yosida regularization of a lower semicontinuous proper convex function is implied by the semismoothness of the metric projector over the epigraph of the convex function.

Two stage equilibrium and product choice with elastic demand

Abstract This paper examines a two stage model of product choice with elastic demand. Duopolists choose locations and then prices. Consumers’ demand is linear in price. Transportation cost is quadratic and is a lump sum. For each pair of locations, there is a price equilibrium in the second stage. The first stage equilibrium locations are unique, symmetric and depend upon the ratio of the reservation price and the transportation cost parameter. When this ratio exceeds a certain critical value, the locations are at the extreme end points of the market. As the ratio decreases, the locations gradually move towards the center.

A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming

In this paper we propose a smoothing Newton-type algorithm for the problem of minimizing a convex quadratic function subject to finitely many convex quadratic inequality constraints. The algorithm is shown to converge globally and possess stronger local superlinear convergence. Preliminary numerical results are also reported.

A Lagrangian Dual Method with Self-Concordant Barriers for Multi-Stage Stochastic Convex Programming

Abstract.This paper presents an algorithm for solving multi-stage stochastic convex nonlinear programs. The algorithm is based on the Lagrangian dual method which relaxes the nonanticipativity constraints, and the barrier function method which enhances the smoothness of the dual objective function so that the Newton search directions can be used. The algorithm is shown to be of global convergence and of polynomial-time complexity.

Interior Point Algorithms For Linear Complementarity Problems Based On Large Neighborhoods Of The Central Path

In this paper we study a first-order and a high-order algorithm for solving linear complementarity problems. These algorithms are implicitly associated with a large neighborhood whose size may depend on the dimension of the problems. The complexity of these algorithms depends on the size of the neighborhood. For the first-order algorithm, we achieve the complexity bound which the typical large-step algorithms possess. It is well known that the complexity of large-step algorithms is greater than that of short-step ones. By using high-order power series (hence the name high-order algorithm), the iteration complexity can be reduced. We show that the complexity upper bound for our high-order algorithms is equal to that for short-step algorithms.

An Analytic Center Cutting Plane Method for Semidefinite Feasibility Problems

Semidefinite feasibility problems arise in many areas of operations research. The abstract form of these problems can be described as finding a point in a nonempty bounded convex body G in the cone of symmetric positive semidefinite matrices. Assume that G is defined by an oracle, which for any givenm xm symmetric positive semidefinite matrix Y either confirms that Y e G or returns a cut, i.e., a symmetric matrixA such that G is in the half-space { Y :A ·Y = A · Y}. We study an analytic center cutting plane algorithm for this problem. At each iteration, the algorithm computes an approximate analytic center of a working set defined by the cutting plane system generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise the new cutting plane returned by the oracle is added into the system. As the number of iterations increases, the working set shrinks and the algorithm eventually finds a solution to the problem. All iterates generated by the algorithm are positive definite matrices. The algorithm has a worst-case complexity of O *( m3 /e 2 )on the total number of cuts to be used, where e is the maximum radius of a ball contained by G.

A Multiple-Cut Analytic Center Cutting Plane Method for Semidefinite Feasibility Problems

We consider the problem of finding a point in a nonempty bounded convex body

Interior-Point Methods with Decomposition for Solving Large-Scale Linear Programs

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