Conic mixed-integer rounding cuts

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce...

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Bibliographic Details
Published in:Mathematical programming Vol. 122; no. 1; pp. 1 - 20
Main Authors: Atamtürk, Alper, Narayanan, Vishnu
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.03.2010
Springer
Springer Nature B.V
Subjects:
Algorithms
Applied sciences
Calculus of Variations and Optimal Control; Optimization
Capital budgeting
Combinatorics
Convex analysis
Exact sciences and technology
Full Length Paper
Integer programming
Linear programming
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Operational research and scientific management
Operational research. Management science
Optimization
Portfolio theory
Software
Studies
Theoretical
Integer programming
Branch-and-cut algorithms
90C57
90C25
90C11
Mixed-integer rounding
Conic programming
Tree(graph)
conic programming
Ordered set
Modeling
Convex programming
Return on investment
Least squares method
Probability learning
Value analysis
Mathematical programming
Branch and bound method
Finance
Linear programming
Mixed integer programming
Cut generation
Stochastic programming
Polyhedron
Branch and cut method
Mixed problem
Solvability
Investment
ISSN:0025-5610, 1436-4646
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Summary:A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean–variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-008-0239-4