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Nonlinear Integer Programming
This chapter is a study of a simple version of general nonlinear integer problems, where all constraints are still linear, and focuses on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure.
How to integrate a polynomial over a simplex
- V. Baldoni, Nicole Berline, J. D. Loera, M. Köppe, M. Vergne
- Mathematics, Computer ScienceMath. Comput.
- 11 September 2008
It is proved that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus, and if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, it is proven that integration can be done inPolynomial time.
Computing Parametric Rational Generating Functions with a Primal Barvinok Algorithm
It is proved that, on the level of indicator functions of polyhedra, there is no need for using inclusion–exclusion formulas to account for boundary effects, and all linear identities in the space of indicator function identities can be purely expressed using partially open variants of the full-dimensionalpolyhedra in the identity.
Software for exact integration of polynomials over polyhedra
- J. D. Loera, B. Dutra, M. Köppe, S. Moreinis, G. Pinto, J. Wu
- Computer Science, MathematicsACCA
- 31 July 2011
A software implementation, part of the software LattE, is described, and benchmark computations are provided for quickly computing the exact value of integrals of polynomial functions over domains that are decomposable into convex polyhedra.
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.
Pareto Optima of Multicriteria Integer Linear Programs
We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered…
Algebraic and Geometric Ideas in the Theory of Discrete Optimization
Algebraic and Geometric Ideas in the Theory of Discrete Optimization offers several research technologies not yet well known among practitioners of discrete optimization, minimizes prerequisites for learning these methods, and provides a transition from linear discrete optimization to nonlinear discrete optimization.
Integer Polynomial Optimization in Fixed Dimension
- J. D. Loera, R. Hemmecke, M. Köppe, R. Weismantel
- Mathematics, Computer ScienceMath. Oper. Res.
- 5 October 2004
For the optimization of an integer polynomial over the lattice points of a convex polytope, an algorithm is shown to compute lower and upper bounds for the optimal value.
On the Complexity of Nonlinear Mixed-Integer Optimization
- M. Köppe
- Computer Science
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…
SBROME: a scalable optimization and module matching framework for automated biosystems design.
- Linh Huynh, A. Tsoukalas, M. Köppe, I. Tagkopoulos
- Computer ScienceACS synthetic biology
- 11 March 2013
A divide-and-conquer Synthetic Biology Reusable Optimization Methodology (SBROME) is proposed, which successfully applied SBROME toward two alternative designs of a modular 3-input multiplexer that utilize pre-existing logic gates and characterized biological parts.