The aim of this paper is to study how efficiently we evaluate a system of multivariate polynomials and their partial derivatives in homotopy continuation methods. Our major tool is an extension of the Hornor scheme, which is popular in evaluating a univariate polynomial, to a multivariate polynomial. But the extension is not unique, and there … Read more
The computation of leastcore and prenucleolus is an efficient way of allocating a common resource among N players. It has, however, the drawback being a linear programming problem with 2^N-2 constraints. In this paper we show how, in the case of convex production games, generate constraints by solving small size linear programming problems, with both … Read more
We present polynomial-time interior-point algorithms for solving the Fisher and Arrow-Debreu competitive market equilibrium problems with linear utilities and $n$ players. Both of them have the arithmetic operation complexity bound of $O(n^4\log(1/\epsilon))$ for computing an $\epsilon$-equilibrium solution. If the problem data are rational numbers and their bit-length is $L$, then the bound to generate an … Read more
We extend the analysis of [27] to handling more general utility functions: piece-wise linear functions, which include Leontief’s utility. We show that the problem reduces to the general analytic center model discussed in [27]. Thus, the same linear programming complexity bound applies to approximating the Fisher equilibrium problem with these utilities. More importantly, we show … Read more
A notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line, when the initial distance apart between the two players is 2. We show that the symmetric rendezvous value is within the interval (4.1520, 4.2574), which considerably improves the previous best … Read more
Korkin and Zolotarev showed that if $$\sum_i A_i(x_i-\sum_{j>i} \alpha_{ij}x_j)^2$$ is the Lagrange expansion of a Korkin–Zolotarev reduced positive definite quadratic form, then $A_{i+1}\geq \frac{3}{4} A_i$ and $A_{i+2}\geq \frac{2}{3}A_i$. They showed that the implied bound $A_{5}\geq \frac{4}{9}A_1$ is not attained by any KZ-reduced form. We propose a method to optimize numerically over the set of Lagrange … Read more
In this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations. Our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size. This is a well-studied problem in case the samples are independent and identically distributed (i.e., when … Read more
One of perspective ways to exploit sparsity in the dependency graph of an optimization problem as J.N. Hooker stressed is nonserial dynamic programming (NSDP) which allows to compute solution in stages, each of them uses results from previous stages. The class of discrete optimization problems with the block-tree-structure matrix of constraints is considered. Nonserial dynamic … Read more
This paper addresses the general single-machine earliness-tardiness problem with distinct release dates, due dates, and unit costs. The aim of this research is to obtain an exact nonpreemptive solution in which machine idle time is allowed. In a hybrid approach, we formulate and then solve the problem using dynamic programming (DP) while incorporating techniques from … Read more
We study a pricing problem where buyers with non-uniform demand purchase one of many items. Each buyer has a known benefit for each item and purchases the item that gives the largest utility, which is defined to be the difference between the benefit and the price of the item. The optimization problem is to decide … Read more