Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Software tools for stochastic programming: A Stochastic Programming Integrated Environment (SPInE)

SP models combine the paradigm of dynamic linear programming with modelling of random parameters, providing optimal decisions which hedge against future uncertainties. Advances in hardware as well as software techniques and solution methods have made SP a viable optimisation tool. We identify a growing need for modelling systems which support the creation and investigation of SP problems. Our SPInE system integrates a number of components which include a flexible modelling tool (based on stochastic extensions of the algebraic modelling languages AMPL and MPL), stochastic solvers, as well as special purpose scenario generators and database tools. We introduce an asset/liability management model and illustrate how SPInE can be used to create and process this model as a multistage SP application

Ethical Stochastic Objectives Programming Approach for Portfolio Selection

The paper develops an ethical multiple stochastic objectives approach to address the ethical portfolio selection problem in the stochastic environment under the Shari’ah compliant framework. Two random objectives considered in this paper which are maximizing portfolio return and maximizing social welfare of portfolio. The risk of portfolio is measured by covariance matrix of total return. The ethical stochastic objectives program approach is based on goal programming approach, a chance constrained approach and Shari’ah compliant framework. The model is applied on 60 stocks including conventional and Islamic securities in GCC. The results show that, portfolios with higher proportion of ethical Islamic securities in the portfolio and with higher expected loss the higher is the portfolio performance in terms of Sharpe measure. Keywords: Shari’ah compliant, Ethical investment, Goal programming, Multiple objectives, Stochastic Multiple objectives programming, Chance constrained approach, Sharpe index as portfolio performance measure

Stochastic Optimization for Financial Decision Making: Portfolio Selection Problem [QA402.5. K45 2008 f rb].

Tesis ini mengaplikasikan pengoptimuman berstokastik sebagai penyelesaian kepada masaalah pemilihan portfolio. Pemilihan portfolio merupakan satu bidang penting dalam pembuatan keputusan kewangan. Ciri penting bagi masaalah dalam pasaran kewangan umumnya terpisah dan tertakrif dengan jelas. In this thesis stochastic optimization was applied to solve portfolio selection problem. Portfolio selection problem is one of the important areas in financial decision making. An important distinguishing feature of problems in financial markets is that they are generally separable and well defined

A Note on the Quantile Formulation

Many investment models in discrete or continuous-time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change-of-variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank-dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton's portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well-posedness, attainability and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law-invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under CPT or RDUT.Comment: to appear in Mathematical Financ

A two-stage stochastic mixed-integer program modelling and hybrid solution approach to portfolio selection problems

In this paper, we investigate a multi-period portfolio selection problem with a comprehensive set of real-world trading constraints as well as market random uncertainty in terms of asset prices. We formulate the problem into a two-stage stochastic mixed-integer program (SMIP) with recourse. The set of constraints is modelled as mixed-integer program, while a set of decision variables to rebalance the portfolio in multiple periods is explicitly introduced as the recourse variables in the second stage of stochastic program. Although the combination of stochastic program and mixed-integer program leads to computational challenges in finding solutions to the problem, the proposed SMIP model provides an insightful and flexible description of the problem. The model also enables the investors to make decisions subject to real-world trading constraints and market uncertainty. To deal with the computational difficulty of the proposed model, a simplification and hybrid solution method is applied in the paper. The simplification method aims to eliminate the difficult constraints in the model, resulting into easier sub-problems compared to the original one. The hybrid method is developed to integrate local search with Branch-and-Bound (B&B) to solve the problem heuristically. We present computational results of the hybrid approach to analyse the performance of the proposed method. The results illustrate that the hybrid method can generate good solutions in a reasonable amount of computational time. We also compare the obtained portfolio values against an index value to illustrate the performance and strengths of the proposed SMIP model. Implications of the model and future work are also discussed