# Convex Discrete Optimization

@inproceedings{Onn2007ConvexDO, title={Convex Discrete Optimization}, author={Shmuel Onn}, booktitle={Encyclopedia of Optimization}, year={2007} }

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering…

## 20 Citations

### Complexity of Degeneracy

- Computer Science, MathematicsEncyclopedia of Optimization
- 2009

Introduction Limitations Outline and Overview of Main Results and Applications Terminology and Complexity Reducing Convex to Linear Discrete Optimization Edge-Directions and Zonotopes Strongly…

### Approximation Algorithms for Reliable Stochastic Combinatorial Optimization

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2010

This paper constructs a δ(1 + e)-approximation algorithm for the stochastic problem, which invokes the linear algorithm only a logarithmic number of times in the problem input, for any desired accuracy level e > 0.

### Graver basis and proximity techniques for block-structured separable convex integer minimization problems

- Computer Science, MathematicsMath. Program.
- 2014

This algorithm combines Graver basis techniques with a proximity result, which allows for the minimization of separable convex objective functions and to use convex continuous optimization as a subroutine.

### Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

- Computer ScienceOberwolfach Reports
- 2015

MINLP is one of the most general classes of MP, which is itself a formal language used to describe optimization problems in terms of parameters, decision variables, an objective function to be optimized, and some constraints to be satisfied.

### On Approximation Algorithms for Concave Mixed-Integer Quadratic Programming

- Mathematics, Computer ScienceIPCO
- 2016

An algorithm that finds an epsilon-approximate solution to a concave mixed-integer quadratic programming problem and is polynomial in the size of the problem and in 1/\epsilon, provided that the number of integer variables and thenumber of negative eigenvalues of the objective function are fixed.

### Theory and Applications of N-Fold Integer Programming

- Computer Science, MathematicsArXiv
- 2009

We overview our recently introduced theory of n-fold integer programming which enables the polynomial time solution of fundamental linear and nonlinear integer programming problems in variable…

### Algorithmic and modeling insights via volumetric comparison of polyhedral relaxations

- MathematicsMath. Program.
- 2018

The motivation is in geometrically comparing relaxations in the context of mixed-integer linear and nonlinear optimization, with the goal of gaining algorithmic and modeling insights.

### On the Complexity of Nonlinear Mixed-Integer Optimization

- Computer Science
- 2012

This is a survey on the computational complexity of nonlinear mixedinteger optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number…

### On the complexity of nonlinear mixed-integer optimization

- Computer Science
- 2010

This is a survey on the computational complexity of nonlinear mixedinteger optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number…

### A note on a selfish bin packing problem

- EconomicsJ. Glob. Optim.
- 2013

A selfish bin packing problem, where each item is a selfish player and wants to minimize its cost, is considered, and it is shown that any feasible packing can be converged to a Nash Equilibrium in O(n2) steps without increasing the social cost.

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