# Polyhedral Outer Approximations in Convex Mixed-Integer Nonlinear Programming

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Abstrakt

This thesis is focused on a specific type of optimization problems

commonly referred to as convex MINLP problems. The goal has been to

investigate and develop efficient methods for solving such optimization

problems. The thesis focuses on decompositionbased algorithms in which a

polyhedral outer approximation is used for approximating the integer

relaxed feasible set. An introduction and summary of different types of

optimization problems are given in Chapter 2, and a summary of work

previously done within the field is given in Chapter 3.

The thesis is written as a collection of articles, and the results

presented in the papers are summarized in Chapter 4. Different

approaches for constructing and utilizing polyhedral outer

approximations are presented and analyzed within the thesis. An

algorithm, called the extended supporting hyperplane (ESH) algorithm,

which uses supporting hyperplanes to the integer relaxed feasible set to

construct the polyhedral approximation is presented. The ESH algorithm

is utilized in a new solver, called SHOT, which is described in the

thesis and presented in detail in the enclosed papers. Different

techniques for utilizing the polyhedral approximations are presented in

the thesis, and numerical test verifies their efficiency. Reformulation

techniques for utilizing separability of the nonlinear functions to

construct tighter approximations are also described and analyzed.