A Path to the Arrow-Debreu Competitive Market Equilibrium

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

Improved bounds for the symmetric rendezvous search problem on the line

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

New Korkin-Zolotarev Inequalities

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

On Rates of Convergence for Stochastic Optimization Problems Under Non-I.I.D. Sampling

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

Nonserial dynamic programming and local decomposition algorithms in discrete programming

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

New solution approaches to the general single machine earliness-tardiness problem

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

A pricing problem under Monge property

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

The Effects of Adding Objectives to an Optimization Problem on the Solution Set

Suppose that for a given optimisation problem (which might be multicriteria problem or a single-criteron problem), an additional objective function is introduced. How does the the set of solutions, i.~e.\ the set of efficient points change when instead of the old problem the new multicriteria problem is considered? How does the set of properly efficient … Read more

Existence of Equilibrium for Integer Allocation Problems

In this paper we show that if all agents are equipped with discrete concave production functions, then a feasible price allocation pair is a market equilibrium if and only if it solves a linear programming problem, similar to, but perhaps simpler than the one invoked in Yang (2001). Using this result, but assuming discrete concave … Read more

The Efficient Outcome Set of a Bi-criteria Linear Programming and Application

We study the efficient outcome set $Y_E$ of a bi-criteria linear programming problem $(BP)$ and present a quite simple algorithm for generating all extreme points of $Y_E$. Application to optimization a scalar function $h(x)$ over the efficient set of $(BP)$ in case of $h$ which is a convex and dependent on the criteria is considered. … Read more