A value estimation approach to Iri-Imai's method for constrained convex optimization.

Lam Sze Wan.Thesis (M.Phil.)--Chinese University of Hong Kong, 2002.Includes bibliographical references (leaves 93-95).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Background --- p.4Chapter 3 --- Review of Iri-Imai Algorithm for Convex Programming Prob- lems --- p.10Chapter 3.1 --- Iri-Imai Algorithm for Convex Programming --- p.11Chapter 3.2 --- Numerical Results --- p.14Chapter 3.2.1 --- Linear Programming Problems --- p.15Chapter 3.2.2 --- Convex Quadratic Programming Problems with Linear Inequality Constraints --- p.17Chapter 3.2.3 --- Convex Quadratic Programming Problems with Con- vex Quadratic Inequality Constraints --- p.18Chapter 3.2.4 --- Summary of Numerical Results --- p.21Chapter 3.3 --- Chapter Summary --- p.22Chapter 4 --- Value Estimation Approach to Iri-Imai Method for Con- strained Optimization --- p.23Chapter 4.1 --- Value Estimation Function Method --- p.24Chapter 4.1.1 --- Formulation and Properties --- p.24Chapter 4.1.2 --- Value Estimation Approach to Iri-Imai Method --- p.33Chapter 4.2 --- "A New Smooth Multiplicative Barrier Function Φθ+,u" --- p.35Chapter 4.2.1 --- Formulation and Properties --- p.35Chapter 4.2.2 --- "Value Estimation Approach to Iri-Imai Method by Us- ing Φθ+,u" --- p.41Chapter 4.3 --- Convergence Analysis --- p.43Chapter 4.4 --- Numerical Results --- p.46Chapter 4.4.1 --- Numerical Results Based on Algorithm 4.1 --- p.46Chapter 4.4.2 --- Numerical Results Based on Algorithm 4.2 --- p.50Chapter 4.4.3 --- Summary of Numerical Results --- p.59Chapter 4.5 --- Chapter Summary --- p.60Chapter 5 --- Extension of Value Estimation Approach to Iri-Imai Method for More General Constrained Optimization --- p.61Chapter 5.1 --- Extension of Iri-Imai Algorithm 3.1 for More General Con- strained Optimization --- p.62Chapter 5.1.1 --- Formulation and Properties --- p.62Chapter 5.1.2 --- Extension of Iri-Imai Algorithm 3.1 --- p.63Chapter 5.2 --- Extension of Value Estimation Approach to Iri-Imai Algo- rithm 4.1 for More General Constrained Optimization --- p.64Chapter 5.2.1 --- Formulation and Properties --- p.64Chapter 5.2.2 --- Value Estimation Approach to Iri-Imai Method --- p.67Chapter 5.3 --- Extension of Value Estimation Approach to Iri-Imai Algo- rithm 4.2 for More General Constrained Optimization --- p.69Chapter 5.3.1 --- Formulation and Properties --- p.69Chapter 5.3.2 --- Value Estimation Approach to Iri-Imai Method --- p.71Chapter 5.4 --- Numerical Results --- p.72Chapter 5.4.1 --- Numerical Results Based on Algorithm 5.1 --- p.73Chapter 5.4.2 --- Numerical Results Based on Algorithm 5.2 --- p.76Chapter 5.4.3 --- Numerical Results Based on Algorithm 5.3 --- p.78Chapter 5.4.4 --- Summary of Numerical Results --- p.86Chapter 5.5 --- Chapter Summary --- p.87Chapter 6 --- Conclusion --- p.88Bibliography --- p.93Chapter A --- Search Directions --- p.96Chapter A.1 --- Newton's Method --- p.97Chapter A.1.1 --- Golden Section Method --- p.99Chapter A.2 --- Gradients and Hessian Matrices --- p.100Chapter A.2.1 --- Gradient of Φθ(x) --- p.100Chapter A.2.2 --- Hessian Matrix of Φθ(x) --- p.101Chapter A.2.3 --- Gradient of Φθ(x) --- p.101Chapter A.2.4 --- Hessian Matrix of φθ (x) --- p.102Chapter A.2.5 --- Gradient and Hessian Matrix of Φθ(x) in Terms of ∇xφθ (x) and∇2xxφθ (x) --- p.102Chapter A.2.6 --- "Gradient of φθ+,u(x)" --- p.102Chapter A.2.7 --- "Hessian Matrix of φθ+,u(x)" --- p.103Chapter A.2.8 --- "Gradient and Hessian Matrix of Φθ+,u(x) in Terms of ∇xφθ+,u(x)and ∇2xxφθ+,u(x)" --- p.103Chapter A.3 --- Newton's Directions --- p.103Chapter A.3.1 --- Newton Direction of Φθ (x) in Terms of ∇xφθ (x) and ∇2xxφθ(x) --- p.104Chapter A.3.2 --- "Newton Direction of Φθ+,u(x) in Terms of ∇xφθ+,u(x) and ∇2xxφθ,u(x)" --- p.104Chapter A.4 --- Feasible Descent Directions for the Minimization Problems (Pθ) and (Pθ+) --- p.105Chapter A.4.1 --- Feasible Descent Direction for the Minimization Prob- lems (Pθ) --- p.105Chapter A.4.2 --- Feasible Descent Direction for the Minimization Prob- lems (Pθ+) --- p.107Chapter B --- Randomly Generated Test Problems for Positive Definite Quadratic Programming --- p.109Chapter B.l --- Convex Quadratic Programming Problems with Linear Con- straints --- p.110Chapter B.l.1 --- General Description of Test Problems --- p.110Chapter B.l.2 --- The Objective Function --- p.112Chapter B.l.3 --- The Linear Constraints --- p.113Chapter B.2 --- Convex Quadratic Programming Problems with Quadratic In- equality Constraints --- p.116Chapter B.2.1 --- The Quadratic Constraints --- p.11

Comparison of the Computational Performance of Three Quadratic Programming Algorithms

The main objective of this study is to compare the computational perforinance of three quadratic programming algorithms. A quadratic programming problem is one in which the objective function to be minimized is quadratic and the constraint functions are linear. The three algorithms are Wolfe's reduced gradient method (implemented in the MINOS package), Lemke's complementary pivot method, and Fletcher's active set method. Fletcher's method was shown to be superior to the other two methods. In this paper, a random-problems generator is used. In addition, a translator program has been written which tranforms a given input data into MPS and SPECS files which are needed for the MINOS package. In a recent study, it was shown that Lemke's algorithm terminated with an infeasible solution in a convex quadratic programming problem. 'Ibis claim was investigated to know the reason for such an abnormal behavior. 'Ibis investigation is a secondary objective of the study.Computing and Information Science