“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

First Year – Fall ......................... .............................................................................................. Credits MAT 210 ..................................... Calculus I ............................................................................. 4.0 ENG 110...................................... Expository Writing ............................................................. 3.0 CHE 210 ...................................... Chemistry I .......................................................................... 4.0 HUM 100 OR SOC 101 OR PSY 101 ........................................................................................ 6.0 Subtotal ...................................... .............................................................................................. 17

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Alfred Weber's Theory of the Location of Industries@@@Coastwise Cruising, from Erith to Lowestoft

In this paper, we present a review of deterministic software for solving convex MINLP problems as well as a comprehensive comparison of a large selection of commonly available solvers. As a test set, we have used all MINLP instances classified as convex in the problem library MINLPLib, resulting in a test set of 335 convex MINLP instances. A summary of the most common methods for solving convex MINLP problems is given to better highlight the differences between the solvers. To show how the solvers perform on problems with different properties, we have divided the test set into subsets based on the continuous relaxation gap, the degree of nonlinearity, and the relative number of discrete variables. The results also provide guidelines on how well suited a specific solver or method is for particular types of MINLP problems.

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A review and comparison of solvers for convex MINLP

This work presents a review of the main deterministic mixed-integer nonlinear programming (MINLP) solution methods for problems with convex and nonconvex functions. An overview for deriving MINLP formulations through generalized disjunctive programming (GDP), which is an alternative higher-level representation of MINLP problems, is also presented. A review of solution methods for GDP problems is provided. Some relevant applications of MINLP and GDP in process systems engineering are described in this work.

Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Methods

Discrete-continuous optimization problems are commonly modeled in algebraic form as mixed-integer linear or nonlinear programming models. Since these models can be formulated in different ways, leading either to solvable or nonsolvable problems, there is a need for a systematic modeling framework that provides a fundamental understanding on the nature of these models. This work presents a modeling framework, generalized disjunctive programming (GDP), which represents problems in terms of Boolean and continuous variables, allowing the representation of constraints as algebraic equations, disjunctions and logic propositions. An overview is provided of major research results that have emerged in this area. Basic concepts are emphasized as well as the major classes of formulations that can be derived. These are illustrated with a number of examples in the area of process systems engineering. As will be shown, GDP provides a structured way for systematically deriving mixed-integer optimization models that exhibit strong continuous relaxations, which often translates into shorter computational times. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3276–3295, 2013

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Systematic modeling of discrete-continuous optimization models through generalized disjunctive programming

http://egon.cheme.cmu.edu/Papers/Trespalacios_new_bm.pdf

Improved Big-M reformulation for generalized disjunctive programs

An MILP-MINLP decomposition method for the global optimization of a source based model of the multiperiod blending problem

The solver DICOPT is based on the outer-approximation algorithm used for solving mixed-integer nonlinear programming (MINLP) problems. This algorithm is very effective for solving some type...

Francisco Trespalacios

Papers

Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Methods

Systematic modeling of discrete-continuous optimization models through generalized disjunctive programming

Improved Big-M reformulation for generalized disjunctive programs

An MILP-MINLP decomposition method for the global optimization of a source based model of the multiperiod blending problem

Improving the performance of DICOPT in convex MINLP problems using a feasibility pump