Projected Perspective Reformulations with Applications in Design Problems

Authors:
Antonio Frangioni;Claudio Gentile;Enrico Grande;Andrea Pacifici
Affiliations:
Dipartimento di Informatica, Università di Pisa, Polo Universitario della Spezia, 19121 La Spezia, Italy;Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”---CNR, 00185 Rome, Italy;Dipartimento di Ingegneria dell'Impresa, Università degli Studi di Roma “Tor Vergata,” 00133 Rome, Italy;Dipartimento di Ingegneria dell'Impresa, Università degli Studi di Roma “Tor Vergata,” 00133 Rome, Italy
Venue:
Operations Research
Year:
2011

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Abstract

The perspective relaxation (PR) is a general approach for constructing tight approximations to mixed-integer nonlinear programs (MINLP) with semicontinuous variables. The PR of a MINLP can be formulated either as a mixed-integer second-order cone program (MI-SOCP), provided that the original objective function is SOCP-representable, or as a semi-infinite MINLP. In this paper, we show that under some further assumptions (rather restrictive, but satisfied in several practical applications), the PR of a mixed-integer quadratic program (MIQP) can also be reformulated as a piecewise-quadratic program (QP), ultimately yielding a QP relaxation of roughly the same size of the standard continuous relaxation. Furthermore, if the original problem has some exploitable structure, then this structure is typically preserved in the reformulation, thus allowing the construction of specialized approaches for solving the PR. We report on implementing these ideas on two MIQPs with appropriate structure: a sensor placement problem and a quadratic-cost (single-commodity) network design problem.