Lecture Notes on Engineering Optimization

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Published Date:14-07-2017
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1 ' Lecture Notes on Engineering Optimization Fraser J. Forbes and Ilyasse Aksikas Department of Chemical and Materials Engineering University of Alberta aksikasualberta.ca & %2 ' Contents Chapter 1 : Introduction - Basic Concepts Chapter 2 : Linear Programming Chapter 3 : Unconstrained Univariate Nonlinear Optimization Chapter 4 : Unconstrained Multivariate Nonlinear Optimization Chapter 5 : Constrained Nonlinear Optimization & %3 ' Chapter I Introduction - Basic Concepts Introductory Example General Formulation and Classi cation Development Stages Degrees of Freedom & %1 INTRODUCTORY EXAMPLE 4 ' 1 Introductory Example Problem Schedule your activities for next week-end, given you only have 100 to spend. Question How can we go about solving this problem in a coherent fashion ? & %1 INTRODUCTORY EXAMPLE 5 ' A systematic approach might be : Step 1 : Choose a scheduling objective. Step 2 : List all possible activities and pertinent information regarding each activity, activity fun units/hr cost time limits F C L i i i & %1 INTRODUCTORY EXAMPLE 6 ' Step 3 : Specify other limitations. t = available = available Step 4 : Decide what the decision variables are. Step 5 : Write down any relationships you know of among the variables. & %1 INTRODUCTORY EXAMPLE 7 ' Step 6 : Write down all restrictions on the variables. Step 7 : Write down the mathematical expression for the objective. & %1 INTRODUCTORY EXAMPLE 8 ' Step 8 : Write the problem out in a standard form. There is a more compact way to write this problem down. matrix-vector form. & %2 GENERAL FORMULATION 9 ' 2 General Formulation Optimize P(x) Objective function Performance function Pro t (cost) function Subject to f(x) = 0 Equality constraints Process model g(x) 0 Inequality constraints Operating constraints x xx Variable bounds l u Variables limits & %3 DEVELOPMENT STAGES 10 ' 3 Development Stages Stage 1 : Problem De nition Objectives goal of optimization performance measurement example : max pro t, min energy cost, etc max(P (x)) =min(P (x)). & %3 DEVELOPMENT STAGES 11 ' Process models assumptions & constants equations to represent relationships ? material / energy balances ? thermodynamics ? kinetics, etc constraints and bounds to represent limitations ? operating restrictions, ? bounds / limits A point which satis es the process model equations is a feasible point. Otherwise, it is an infeasible point. & %3 DEVELOPMENT STAGES 12 ' Stage 2 : Problem Formulation Standard form Degrees of freedom analysis over-, under- or exactly speci ed ? decision variables (independent vs. dependent) Scaling of variables units of measure scaling factors & %3 DEVELOPMENT STAGES 13 ' Stage 3 : Problem Solution Technique selection matched to problem type exploit problem structure knowledge of algorithms strengths & weaknesses Starting points usually several and compare Algorithm tuning termination criteria convergence tolerances step length & %3 DEVELOPMENT STAGES 14 ' Stage 4 : Results Analysis Solution uniqueness Perturbation of optimization problem e ects of assumptions variation (uncertainty) in problem parameters variation in prices/costs & %3 DEVELOPMENT STAGES 15 ' Solution of an optimization problem requires all of the steps a full understanding is developed by following the complete cycle good decisions require a full understanding of the problem shortcuts lead to bad decisions. & %3 DEVELOPMENT STAGES 16 ' & %3 DEVELOPMENT STAGES 17 ' In this course, our problems will have : - objective functions which are continuously di erentiable, - constraint equations which are continuously di erentiable. Optimization problems with scalar objective functions and vector constraints can be classi ed : & %4 DEGREES OF FREEDOM 18 ' 4 Degrees of Freedom To determine the degrees of freedom (the number of variables whose values may be independently speci ed) in our model we could simply count the number of independent variables (the number of variables which remain on the right- hand side) in our modi ed equations. This suggests a possible de nition : degrees of freedom = variables equations De nition : The degrees of freedom for a given problem are the number of independent problem variables which must be speci ed to uniquely determine a solution. & %4 DEGREES OF FREEDOM 19 ' Consider the following three equations relating three variables x ;x and x : 1 2 3 x x = 0 1 2 x 2x = 0 1 3 x 2x = 0 2 3 This seems to indicate that there are no degrees of freedom. Notice that if we subtract the last from the second equation : x 2x = 0 1 3 x 2x = 0 2 3 x x = 0 1 2 the result is the rst equation. It seems that we have three di erent equations, which contain no more information than two of the equations. In fact any of the equations is a linear combination of the other two equations. We require a clearer, more precise de nition for degrees of freedom. & %4 DEGREES OF FREEDOM 20 ' A More Formal Approach : Suppose we have a set of m equations : h(v) = 0 in the set of variables v (n +m elements). We would like to determine whether the set of equations can be used to solve for some of the variables in terms of the others. In this case we have a system of m equations in n +m unknown variables. The Implicit Function Theorem states that if the m equations are linearly independent, then we can divide our set of variables v into m dependent variables u and n independent variables x : The Implicit Function Theorem goes on to give conditions under which the dependent variables u may be expressed in terms of the independent variables x or : u =g(x) & %

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