Optimization modeling for portfolio selection and Boolean network problems

Abstract

This thesis focuses on mathematical modeling and optimization for addressing practical portfolio selection and network problems. Portfolio selection problems include first-order and high-order Markov-switching portfolio selection, and online portfolio selection. Network problems include GPS trajectory data segmentation, distribution of successors and discrimination of attractors in Boolean networks. Fuzzy financial portfolio selection problem considering capital gain tax is studied. The Markov-switching mean-variance model is constructed by assuming that the market state switches constantly. The time varying numerical integral-based particle swarm optimization (TVNIPSO) algorithm is designed to obtain the efficient frontier. Numerical experiments are provided to validate the effectiveness of the algorithm. High-order Markov transition process can better depict the market state changes and incorporate more market information due to the financial market has the long memory property. Therefore, the high-order Markov-switching portfolio selection model (HOMSPSM) is proposed where the capital gain-loss offsetting is studied explicitly. The Monte Carlo simulation-based particle swarm optimization (MCPSO) algorithm is designed to solve the proposed model. Numerical experiments demonstrate the strong practicability of HOMSPSM and MCPSO. Since the market information is constantly updated, online portfolio selection problem is studied where the investment strategies are adjusted frequently. The adaptive online moving average (AOLMA) method is proposed to predict the future returns of risky assets and the net profit maximization (NPM) model is constructed. The adaptive online net profit maximization (AOLNPM) algorithm is designed by integrating AOLMA and NPM. Numerical experiments show that AOLNPM dominates multiple state-of-the-art online portfolio selection algorithms. To mine business information from the trajectory data of logistics vehicles, the probabilistic logic-based data segmentation method (PLDSM) is proposed by analysing the driving habits, stopping points and duration of nonmovement. An efficient numerical algorithm integrating duality theory and Newton's method is designed. The results show that the PLDSM not only helps finding the business points but also assists in inferring the business affair categories, which enrich the data segmentation technique and promote the practicability of probabilistic logic. The thesis also studies how entropy changes via the transition in Boolean networks (BNs), including BNs consisting of exclusive OR (XOR) functions, canalyzing functions, and threshold functions. The results show that there exists a BN consisting of d-ary XOR functions which preserves the entropy if d is odd and n>d (n is the dimension of the input), whereas there does not exist such a BN if d is even. There exists a BN consisting of d-ary threshold functions which preserves the entropy if n mod d = 0. Furthermore, the upper and lower bounds of the entropy for BNs consisting of canalyzing functions are theoretically analyzed. Discrimination of attractors is crucial to observe the internal state of large-scale networks, this thesis focuses on determining the minimum number of nodes to discriminate singleton attractors under the assumption that each attractor has at most K noisy nodes. The exact and approximation algorithms are designed, and their effectiveness is validated in the numerical experiments.