An interior-point and decomposition approach to multiple stage stochastic programming

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Global error bounds for convex conic problems

In this paper Lipschitzian type error bounds are derived for general convex conic problems under various regularity conditions. Specifically, it is shown that if the recession directions satisfy Slater's condition then a global Lipschitzian type error bound holds. Alternatively, if the feasible region is bounded, then the ordinary Slater condition guarantees a global Lipschitzian type error bound. These can be considered as generalizations of previously known results for inequality systems. Moreover, some of the results are also generalized to the intersection of multiple cones. Under Slater's condition alone, a global Lipschitzian type error bound may not hold. However, it is shown that such an error bound holds for a specific region. For linear systems we show that the constant involved in Hoffman's error bound can be estimated by the so-called condition number for linear programming

A note on a profit maximizing location model

In this paper we discuss a locational model with a profit-maximizing objective. The model can be illustrated by the following situation. There is a set of potential customers in a given region. A firm enters the market and wants to sell a certain product to this set of customers. The location and demand of each potential customer are assumed to be known. In order to maximize its total profit, the firm has to decide: 1) where to locate its distribution warehouse to serve the customers; 2) the price for its product. Due to existence of competition, each customer holds a reservation price for the product. This reservation price is a decreasing function in the distance to the warehouse. If the actual price is higher than the reservation price, then the customer will turn to some other supplier and hence is lost from the firm's market. The problem of the firm is to find the best location for its warehouse and the best price for its product at the same time in order to maximize the total profit. We show that this problem can be solved in polynomial time

On Purchase Timing Models in Marketing

In this paper we consider stochastic purchase timing models used in marketing for low-involvement products and show that important characteristics of those models are easy to compute. As such these calculations are based on an elementary probabilistic argument and cover not only the well-known condensed negative binomial model but also stochastic purchase timing models with other interarrival and mixing distributions

Analytic central path, sensitivity analysis and parametric linear programming

In this paper we consider properties of the central path and the analytic center of the optimal face in the context of parametric linear programming. We first show that if the right-hand side vector of a standard linear program is perturbed, then the analytic center of the optimal face is one-side differentiable with respect to the perturbation parameter. In that case we also show that the whole analytic central path shifts in a uniform fashion. When the objective vector is perturbed, we show that the last part of the analytic central path is tangent to a central path defined on the optimal face of the original problem

Duality Results for Conic Convex Programming

This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone in finite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are given to illustrate these new properties. The topics covered in this paper include Gordon-Stiemke type theorems, Farkas type theorems, perfect duality, Slater condition, regularization, Ramana's duality, and approximate dualities. The dual representations of various convex sets, convex cones and conic convex programs are also discussed

Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming

This paper establishes the superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming under the assumptions that the semidefinite program has a strictly complementary primal-dual optimal solution and that the size of the central path neighborhood tends to zero. The interior point algorithm considered here closely resembles the Mizuno-Todd-Ye predictor-corrector method for linear programming which is known to be quadratically convergent. It is shown that when the iterates are well centered, the duality gap is reduced superlinearly after each predictor step. Indeed, if each predictor step is succeeded by [TeX: $r$] consecutive corrector steps then the predictor reduces the duality gap superlinearly with order [TeX: $\\frac{2}{1+2^{-2r}}$]. The proof relies on a careful analysis of the central path for semidefinite programming. It is shown that under the strict complementarity assumption, the primal-dual central path converges to the analytic center of the primal-dual optimal solution set, and the distance from any point on the central path to this analytic center is bounded by the duality gap

Polynomial Primal-Dual Cone Affine Scaling for Semidefinite Programming

In this paper we generalize the primal--dual cone affine scaling algorithm of Sturm and Zhang to semidefinite programming. We show in this paper that the underlying ideas of the cone affine scaling algorithm can be naturely applied to semidefinite programming, resulting in a new algorithm. Compared to other primal--dual affine scaling algorithms for semidefinite programming, our algorithm enjoys the lowest computational complexity

Conic convex programming and self-dual embedding

How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the two-phase approach or using the so-called big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem, viz. the self-dual embedding technique proposed by Ye, Todd and Mizuno (30). For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving its embedded self-dual problem. The embedded self-dual problem has a trivial initial solution and has the same structure as the original problem. Hence, it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programming. Since a nonlinear conic convex programming problem may lack the so-called strict complementarity property, it causes difficulties in identifying solutions for the original problem, based on solutions for the embedded self-dual system. We provide numerous examples from semidefinite programming to illustrate various possibilities which have no analogue in the linear programming case