Modelling and solution methods for portfolio optimisation

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University, 16/01/2004.In this thesis modelling and solution methods for portfolio optimisation are presented. The investigations reported in this thesis extend the Markowitz mean-variance model to the domain of quadratic mixed integer programming (QMIP) models which are 'NP-hard' discrete optimisation problems. In addition to the modelling extensions a number of challenging aspects of solution algorithms are considered. The relative performances of sparse simplex (SSX) as well as the interior point method (IPM) are studied in detail. In particular, the roles of 'warmstart' and dual simplex are highlighted as applied to the construction of the efficient frontier which requires processing a family of problems; that is, the portfolio planning model stated in a parametric form. The method of solving QMIP models using the branch and bound algorithm is first developed; this is followed up by heuristics which improve the performance of the (discrete) solution algorithm. Some properties of the efficient frontier with discrete constraints are considered and a method of computing the discrete efficient frontier (DEF) efficiently is proposed. The computational investigation considers the efficiency and effectiveness in respect of the scale up properties of the proposed algorithm. The extensions of the real world models and the proposed solution algorithms make contribution as new knowledge

Optimization and Applications

Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research

Modelling and solution methods for portfolio optimisation

In this thesis modelling and solution methods for portfolio optimisation are presented. The investigations reported in this thesis extend the Markowitz mean-variance model to the domain of quadratic mixed integer programming (QMIP) models which are 'NP-hard' discrete optimisation problems. In addition to the modelling extensions a number of challenging aspects of solution algorithms are considered. The relative performances of sparse simplex (SSX) as well as the interior point method (IPM) are studied in detail. In particular, the roles of 'warmstart' and dual simplex are highlighted as applied to the construction of the efficient frontier which requires processing a family of problems; that is, the portfolio planning model stated in a parametric form. The method of solving QMIP models using the branch and bound algorithm is first developed; this is followed up by heuristics which improve the performance of the (discrete) solution algorithm. Some properties of the efficient frontier with discrete constraints are considered and a method of computing the discrete efficient frontier (DEF) efficiently is proposed. The computational investigation considers the efficiency and effectiveness in respect of the scale up properties of the proposed algorithm. The extensions of the real world models and the proposed solution algorithms make contribution as new knowledge.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

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

Scalable Semidefinite Programming

Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This paper develops a provably correct algorithm for solving large SDP problems by economizing on both the storage and the arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment. Running on a laptop, the algorithm can handle SDP instances where the matrix variable has over 10¹³ entries

Interior point based optimal voltage stability, oscillatory stability and ATC margin boundary tracing

This dissertation proposes a general framework for the power system stability margin boundary tracing and optimization. The proposed framework combines interior point algorithm and continuation method seamlessly to provide the optimal control configuration for any feasible system margin. The maximum stability margin for any given control configuration can be derived by interior point based optimal margin boundary tracing (IP-OMBT) while minimizing the corresponding control costs. From the first stability margin boundary point to the maximum margin boundary point, a series of margin levels with corresponding minimal control cost structure are generated. The margin benefit and the corresponding optimal control costs are visualized along the margin boundary. The proposed method is flexible enough to be modified to trace various other security margin boundaries. In addition, direct ATC tracing and optimal ATC tracing package are developed to address voltage/oscillatory stability related ATC problems. Numerical examples with New England 39 buses system are presented to demonstrate the versatility and practical usefulness of IP-OMBT package

Supply function equilibrium analysis for electricity markets

The research presented in this Thesis investigates the strategic behaviour of generating firms in bid-based electricity pool markets and the effects of control methods and network features on the electricity market outcome by utilising the AC network model to represent the electric grid. A market equilibrium algorithm has been implemented to represent the bi-level market problem for social welfare maximization from the system operator and utility assets optimisation from the strategic market participants, based on the primal-dual interior point method. The strategic interactions in the market are modelled using supply function equilibrium theory and the optimum strategies are determined by parameterization of the marginal cost functions of the generating units. The AC power network model explicitly represents the active and reactive power flows and various network components and control functions. The market analysis examines the relation between market power and AC networks, while the different parameterization methods for the supply function bids are also investigated. The first part of the market analysis focuses on the effects of particular characteristics of the AC network on the interactions between the strategic generating firms, which directly affect the electricity market outcome. In particular, the examined topics include the impact of transformer tap-ratio control, reactive power control, different locations for a new entry’s generating unit in the system, and introduction of photovoltaic solar power production in the pool market by considering its dependencyon the applied solar irradiance. The observations on the numerical results have shown that their impact on the market is significant and the employment of AC network representation is required for reliable market outcome predictions and for a better understanding of the strategic behaviour as it depends on the topology of the system. The analysis that examines the supply function parameterizations has shown that the resulting market solutions from the different parameterization methods can be very similar or differ substantially, depending on the presence and level of network congestion and on the size and complexity of the examined system. Furthermore, the convergence performance of the implemented market algorithm has been examined and proven to exhibit superior computational efficiency, being able to provide market solutions for large complex AC systems with multiple asymmetric firms, providing the opportunity for applications on practical electricity markets