Rappaport D., A convex hull algorithm for discs, and applications, Computational Geometry: Theory and Applications 1 (1992) 171-187. We show that the convex hull of a set of discs can be determined in @(n log n) time. The algorithm is straightforward and simple to implement. We then show that the convex hull can be used to efficiently solve various problems for a set of discs. This includes O(n...

We consider the problem of unconstrained minimization of a smooth function in the derivativefree setting using. In particular, we propose and study a simplified variant of the direct search method (of direction type), which we call simplified direct search (SDS). Unlike standard direct search methods, which depend on a large number of parameters that need to be tuned, SDS depends on a single sc...

Given a separation oracle SO for convex function f defined on ℝ n that has an integral minimizer inside box with radius R , we show how to find exact of using at most O(n (n log (n)/ (n) + ( ))) calls and poly )) arithmetic operations, or (nR) exp O(n) ) ⋅ (log (R) operations. When the set minimizers extreme points, our algorithm outputs . This improves upon previously best complexity 2 polynom...

We present a unified analysis for single-machine scheduling problems in which the actual job processing times are controlled by either a linear or a convex resource allocation function and also vary concurrently depending on either the job’s position in the sequence and/or on the total processing time of the already processed jobs. We show that the problem is solvable in Oðn log nÞ time by usin...

The Birkhoff polytope (the convex hull of the set of permutation matrices), which is represented using Θ(n) variables and constraints, is frequently invoked in formulating relaxations of optimization problems over permutations. Using a recent construction of Goemans [1], we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of...

Many applications in the area of production and statistical estimation are problems of convex optimization subject to ranking constraints that represent a given partial order. This problem – which we call the convex cost closure problem, or (CCC) – is a generalization of the known maximum (or minimum) closure problem and the isotonic regression problem. For a (CCC) problem on n variables and m ...