Date of Award

5-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Member

Dr. Warren Adams, Committee Chair

Committee Member

Dr. Xuhong Gao

Committee Member

Dr. Matthew Saltzman

Committee Member

Dr. Cole Smith

Abstract

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.

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