14 research outputs found
Convex envelopes of bounded monomials on two-variable cones
We consider an -variate monomial function that is restricted both in value
by lower and upper bounds and in domain by two homogeneous linear inequalities.
Such functions are building blocks of several problems found in practical
applications, and that fall under the class of Mixed Integer Nonlinear
Optimization. We show that the upper envelope of the function in the given
domain, for is given by a conic inequality. We also present the lower
envelope for . To assess the applicability of branching rules based on
homogeneous linear inequalities, we also derive the volume of the convex hull
for .Comment: 22 pages, 12 figure
Recursive McCormick Linearization of Multilinear Programs
Linear programming (LP) relaxations are widely employed in exact solution
methods for multilinear programs (MLP). One example is the family of Recursive
McCormick Linearization (RML) strategies, where bilinear products are
substituted for artificial variables, which deliver a relaxation of the
original problem when introduced together with concave and convex envelopes. In
this article, we introduce the first systematic approach for identifying RMLs,
in which we focus on the identification of linear relaxation with a small
number of artificial variables and with strong LP bounds. We present a novel
mechanism for representing all the possible RMLs, which we use to design an
exact mixed-integer programming (MIP) formulation for the identification of
minimum-size RMLs; we show that this problem is NP-hard in general, whereas a
special case is fixed-parameter tractable. Moreover, we explore structural
properties of our formulation to derive an exact MIP model that identifies RMLs
of a given size with the best possible relaxation bound is optimal. Our
numerical results on a collection of benchmarks indicate that our algorithms
outperform the RML strategy implemented in state-of-the-art global optimization
solvers.Comment: 22 pages, 11 figures, Under Revie
Robust Assignment of Natural Frequencies and Antiresonances in Vibrating Systems through Dynamic Structural Modification
This paper proposes a novel method for the robust partial assignment of natural frequencies and antiresonances, together with the partial assignment of the related eigenvectors, in lightly damped linear vibrating systems. Dynamic structural modification is exploited to assign the eigenvalues, either of the system or of the adjoint system, together with their sensitivity with respect to some parameters of interest. To handle with constraints on the feasible modifications, the inverse eigenvalue problem is cast as a minimization problem and a solution method is proposed through homotopy optimization. Variables lifting for bilinear and trilinear terms, together with bilinear and double-McCormick's constraints, are exploited to provide a convexification of the problem and to boost the attainment of the global optimum. The effectiveness of the proposed method is assessed through four numerical examples
Error bounds for monomial convexification in polynomial optimization
Convex hulls of monomials have been widely studied in the literature, and
monomial convexifications are implemented in global optimization software for
relaxing polynomials. However, there has been no study of the error in the
global optimum from such approaches. We give bounds on the worst-case error for
convexifying a monomial over subsets of . This implies additive error
bounds for relaxing a polynomial optimization problem by convexifying each
monomial separately. Our main error bounds depend primarily on the degree of
the monomial, making them easy to compute. Since monomial convexification
studies depend on the bounds on the associated variables, in the second part,
we conduct an error analysis for a multilinear monomial over two different
types of box constraints. As part of this analysis, we also derive the convex
hull of a multilinear monomial over .Comment: 33 pages, 2 figures, to appear in journa