Convex Discrete Optimization

  title={Convex Discrete Optimization},
  author={Shmuel Onn},
  booktitle={Encyclopedia of Optimization},
  • S. Onn
  • Published in Encyclopedia of Optimization 1 March 2007
  • Computer Science, Mathematics
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… 

Complexity of Degeneracy

  • K. G. Murty
  • Computer Science, Mathematics
    Encyclopedia 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

  • E. Nikolova
  • Mathematics, Computer Science
  • 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

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

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

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

  • S. Onn
  • Computer Science, Mathematics
  • 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

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

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

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

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.



An application of simultaneous diophantine approximation in combinatorial optimization

A preprocessing algorithm is presented to make certain polynomial time algorithms strongly polynomially bounded in the size of the combinatorial structure and which yields the same set of optimal solutions asw.

N-fold integer programming

A Polynomial Time Algorithm for Shaped Partition Problems

It is shown that when both d and p are fixed, the number of vertices of any shaped partition polytope is O(n^{d{p\choose 2}})$ and all vertices can be produced in strongly polynomial time.

Theory of linear and integer programming

  • A. Schrijver
  • Mathematics
    Wiley-Interscience series in discrete mathematics and optimization
  • 1999
Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear

Algorithmic theory of numbers, graphs and convexity

  • L. Lovász
  • Mathematics
    CBMS-NSF regional conference series in applied mathematics
  • 1986
How to Round Numbers Preliminaries and some Applications in Combinatorics Cuts and Joins Chromatic Number, Cliques and Perfect Graphs Minimizing a Submodular Function.

On the foundations of linear and integer linear programming I

This paper gives a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms, and proves that any “improvement process” which searches through a test set at each stage converges to an optimal point in a finite number of steps.

The Complexity of Generic Primal Algorithms for Solving General Integer Programs

It is shown that one can solve every integer programming problem in polynomial time provided one can efficiently solve the directed augmentation problem, and implies that directed augmentmentation is at least as hard as optimization.

The Vector Partition Problem for Convex Objective Functions

This article shows that when both p,d are fixed, the partition problem is solvable in strongly polynomial time using O(n d(p-1)-1) arithmetic operations, which improves upon the previously known bound of O( ndp 2 ).

An Adaptive Algorithm for Vector Partitioning

An adaptive algorithm for the vector partition problem that runs in time O(q(L)ċv) and in space O(L), where q is a polynomial function, L is the input size and v is the number of vertices of the associated partition polytope, based on an output-sensitive algorithm for enumerating all vertices.

Maximizing Classes of Two-Parameter Objectives Over Matroids

This work finds in strongly polynomial time the minimal cost reliability ratio spanning tree of an undirected graph.