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Signomial Programs with Equality Constraints

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Signomial Programs with Equality Constraints SIGNOMIAL PROGRAMS WITH EQUALITY CONSTRAINTS: NUMERICAL SOLUTION AND APPLICATIONS by JAMES YAN B.A.Sc., University of British Columbia, Vancouver, 1969 -M.A.Sc., University of British Columbia, Vancouver, 1971 •A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Electrical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1976 © James Yan, 1976 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Depa rtment The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT Many design problems from diverse engineering disciplines can be formulated as signomial programs with both equality and inequa• lity constraints. However, existing computational methods of signomial programming are applicable to programs with inequality constraints only. In this thesis an algorithm is proposed and implemented to solve signo• mial programs with mixed inequality and equality constraints. The algo• rithm requires no manipulation of the constraints by the user and is compatible with the results of signomial programming. The proposed algorithm is a synthesis shaped by three concepts: the method of multipliers viewed as a primal-dual method, partial dua- lization,.and the retention of the structure of a signomial program. The strategy of the algorithm is to replace the original problem with a se• quence of subproblems each subject to the original signomial inequality constraints only. The subproblem's objective function is the original cost augmented with the equality constraints and specified by a parameter vector X_ and a penalty constant K. The algorithm then alternates between solving the subproblem and updating X_ and K. The convergence of the algorithm is, under suitable assumptions, guaranteed by the convergence of the method of multipliers and the method used to solve the subproblem. Because the subproblem has the form of a regular signomial program, it can in principle be solved by the techniques of signomial programming. A new numerical method implementing a variant of the Avriel- Williams algorithm for signomial programs is suggested for solving the subproblem. The method relies on monomial condensation, and the combined use of the reduced gradient method and a cutting plane scheme. Details ii of the method's implementation are considered, and some computational experience has been acquired. The proposed method also has the advan• tageous flexibility of being able to handle non-signomial differentiable objective functions. Four updating schemes for X_ and K are formulated and evaluated in a series of numerical experiments. In terms of the rate of convergence, the most promising scheme tested is the use of the Hestenes-Powell rule for updating A_ and the moderate monotonic increase of K after the comple• tion of each subproblem. Convergence can also be considerably accele• rated by properly scaling the equality constraints and performing only inexact minimization in the first few subproblems. The applicability of the algorithms developed in this thesis is illustrated with the solution of three design examples drawn from structural design and chemical process engineering. iii TABLE OF CONTENTS Page ABSTRACT ii TABLE OF CONTENTS ............ iv LIST OF ILLUSTRATIONS vl LIST OF TABLES . vii ACKNOWLEDGEMENT , viii . I. INTRODUCTION 1 1.1 Nonlinear Programming and Engineering Design 1 1.2 A Preview of the Thesis 2 II. SIGNOMIAL PROGRAMMING 6 2.1 Introduction" . 6 2.2 Posynomlals and Their Condensation ... ^ 2.3 Signomial Programming 9 2.4 Geometric Programming H 2.5 Solution of Geometric Programs 2.6 Solution of Signomial Programs 15 III. SIGNOMIAL PROGRAMS WITH EQUALITY CONSTRAINTS . 19 3.1 Motivation i9 3.2 Previous Work 20 3.3 Problem Statement 21 3.4 Proposed Algorithm 22 3.4.1 Background . ; 22 3.4.2 Development of the Algorithm 23 3.4.3 Statement.of the Algorithm . 27 3.5 Convergence Considerations 29 3.6 Solution of the Subproblem 31 3.7 Conclusion 37 IV. A PROPOSED ALGORITHM FOR INEQUALITY-CONSTRAINED SIGNOMIAL PROGRAMS 39 4.1 Introduction . ^9 4.2 Problem Statement 4° 4.3 Proposed Algorithm 41 4.3.1 Preview 41 4.3.2 The Reduced Gradient Method . 41 4.3.3 Algorithm Development 43 4.3.4 The Algorithm 46 4.3.5 The Algorithm in Perspective 46 4.4 Considerations for Implementation 48 4.4.1 Finding a Feasible Point of a Signomial Program. 48 4.4.2 Getting an Initial Basic Feasible Solution of Problem (LST) - No Cut Added .......... 49 4.4.3 Getting an Initial Basic Feasible Solution of Problem (LST) - After the Addition of a Cut . 51 iv Page 9 4.4.4 A Linear Search Method ...... 53 4.4.5 Optimality Criteria, Feasibility Tolerances, and Updating the Basis' Inverse ........ 56 4.5 Computational Experience ............... 58 4.5.1 Posynomial Representation . 58 4.5.2 Software Implementation 59 4.5.3 Computational Results 60 4.6 Presence of Simple Upper Bounds 66 4.7 Minimizing Algebraic Functionals of Signomials .... 70 4.7.1 Introduction ....... 70 4.7.2 Problem Formulation and Solution 7i 4.7.3 Application to Constrained Location-Allocation Problems 72 V. NUMERICAL SOLUTION OF SIGNOMIAL PROGRAMS WITH EQUALITY CONSTRAINTS . 77 77 5.1 Introduction 5.2 Solution of the Subproblem . 78 5.3 Schemes for Updating X_ and K 79 5.3.1 Introduction 79 5.3.2 Choice of Step Size a 81 5.3.3 Updating the Penalty Constant K ........ 84 5.4 Numerical Experiments ^ 5.4.1 Introduction 85 5.4.2 Software Details ........... 85 5.4.3 Test Problems .......... 87 5.4.4 Experiment Plan . 90 no 5.4.5 Results and Discussion ^ VI. APPLICATIONS IN ENGINEERING DESIGN . 100 6.1 Optimum Design of Statically Indeterminate Pin-Jointed Structures . 100 6.1.1 Introduction ..... 100 6.1.2 Problem Formulation 100 6.1.3 Example 103 6.2 Optimization Examples from Chemical Process Engineering 107 6.2.1 Introduction 107 6.2.2 Alkylation Process Optimization 108 6.2.3 Design of a Heat Exchanger Network H6 VII. CONCLUSION . 124 7.1 Synopsis' . 124 7.2 Contributions of the Thesis 128 7.3 Suggestions for Further Research 129 REFERENCES 131 v LIST OF ILLUSTRATIONS Figure Page 5.1 Typical plot of log 6(N) against the dual iteration count N , 97 6.1 The hyperstatic pin-jointed structure of the example, (a) Frame and loading (b) Basic frame with external load (c) Basic frame subject to redundant force .... 104 6.2 Simplified alkylation process diagram . 109 6.3 A small heat exchanger network 117 vi LIST OF TABLES Table Page 4.1 Coefficients for Problem C ......... 63 4.2 Statistics of the test problems for the CRGCP algorithm. 63 '4.3 Starting point and solution of (a) Problem A, (b) Pro• blem B, (c) Problem C . 64 4.4 Coefficients of example problem with simple upper bounds. 69 4.5 Starting points and solutions of the example problem with simple upper bounds . 69 4.6 Locations and weights for major Ukraine cities ..... 76 4.7 Summary of the solution of the constrained location pro• blem 76 5.1 Summary of the characteristics of the test problems . 87 5.2 Performance results of Schemes I-IV 93 6.1 The alkylation process problem's variables and their bounds 110 6.2 Definitions and values of the cost coefficients of the alkylation process problem ..... 110 6.3 The initial and optimal solution of the alkylation process problem iX5 6.4 Data for the heat exchanger network design problem . 121 6.5 Starting point and optimal solution of the heat ex• changer network problem 123 vii ACKNOWLEDGEMENT I would like to thank Dr. Erik V. Bohn for introducing me to geometric programming and for supervising my doctoral research. My first encounter with the method of multipliers was due to Dr. Mordecai Avriel of Technion, Israel. I am also grateful to Dr. Avriel for his encouragement and for the many valuable discussions we had during his stay at U.B.C. in 1974-75. I wish to acknowledge the use of Dr. Ron Dembo's code GGP made available to me by Dr. Avriel. The timely assistance of my friend, Rose Cam, during the pre• paration of the first draft provided a much needed boost. She deserves a bouquet of roses. Many thanks go to Misses Mary-Ellen Flanagan and Sannifer Louie for their excellent work in typing the final copy. Finally, the graduate fellowships from UBC and a National Research Council of Canada Postgraduate Scholarship are gratefully acknowledged. Partial support of the project also came from NRC Grant No. A3134. viii 1. I. INTRODUCTION 1.1 Nonlinear Programming and Engineering Design Design is the essential task of engineering. The design pro• cess usually begins with the recognition of a need. The process then proceeds with the analysis of the need, the synthesis of solution con• cepts likely to satisfy the need, the evaluation of the proposed solu• tions, and the selection of a. final proposal. The selected concept is then further refined and made sufficiently specific for prototype eva• luation and possible revision. Finally, the design plan is implemented. In most stages of the design process opportunities for optimi• zation exist. However, the disparate problems faced in each stage dic• tate the use of radically different optimization methods.
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