Date of Award
Doctor of Philosophy (PhD)
Dr. Warren Adams, Committee Chair
Dr. Xuhong Gao
Dr. Matthew Saltzman
Dr. Cole Smith
Motivated by a variety of problems in global optimization and integer programming that involve multilinear expressions of discrete or continuous variables, this research derives approxima-tions of multilinear functions, and studies the accuracy of these approximations through worst-case error-analyses:
• The derivation of the convex hull representations of large families of symmetric multilinear polynomials (SMPs) that are deﬁned over box constraints through geometrical exploitation of the polytope symmetry and specially designed facet generation method; and
• The identiﬁcation of the set of all points at which a nonnegative multilinear polynomial on a box vanishes, which applies to the identiﬁcation of the set of all points which satisfy any facet at equality.
• The worst-case error analysis associated with linearizations of monomial expressions in boun-ded discrete and/or continuous variables: for certain families of variable-bound structures, the worst-case errors associated with convex hull forms are studied, along with the identiﬁcation of all points which produce these errors.
• The worst-case error analysis associated with replacing the multilinear monomial term with a “best” approximating linear function, in contrast to the previous item on “convex hull linearization:” using the results of the ﬁrst item, explicit convex hull forms are exploited to identify the “best” linear functions.
Xu, Yibo, "Convex Hulls, Relaxations, and Approximations of General Monomials and Multilinear Functions" (2018). All Dissertations. 2094.