Twenty years of linear programming based portfolio optimization

a b s t r a c t Markowitz formulated the portfolio optimization problem through two criteria: the expected return and the risk, as a measure of the variability of the return. The classical Markowitz model uses the variance as the risk measure and is a quadratic programming problem. Many attempts have been made to linearize the portfolio optimization problem. Several different risk measures have been proposed which are computationally attractive as (for discrete random variables) they give rise to linear programming (LP) problems. About twenty years ago, the mean absolute deviation (MAD) model drew a lot of attention resulting in much research and speeding up development of other LP models. Further, the LP models based on the conditional value at risk (CVaR) have a great impact on new developments in portfolio optimization during the first decade of the 21st century. The LP solvability may become relevant for real-life decisions when portfolios have to meet side constraints and take into account transaction costs or when large size instances have to be solved. In this paper we review the variety of LP solvable portfolio optimization models presented in the literature, the real features that have been modeled and the solution approaches to the resulting models, in most of the cases mixed integer linear programming (MILP) models. We also discuss the impact of the inclusion of the real features

A Portfolio Optimization Model

RÉSUMÉ : Ce mémoire étudie le problème d’optimisation de portefeuille moyenne-variance (MV) de Markowitz avec contrainte de cardinalité et bornes sur les variables. C’est un problème NP-Difficile modélisé à l’aide d’un programme MIQP. La performance du portefeuille MV optimal généré est évaluée à l’aide de la méthode exacte Branch-and-Bound (BB) qui fournit une solution optimale globale en comparaison avec les méthodes les plus performantes de la littérature, comme l'écart absolu moyen, la différence moyenne de Gini et la valeur conditionnelle à risque. Ces méthodes alternatives utilisent différentes mesures de risque. De plus, nous avons appliqué pour la première fois une approximation externe (Outer Approximation - OA) au problème MV et nous avons proposé une nouvelle heuristique de branchement afin de faire face à la difficulté de résolution des grandes instances du problème. Ces dernières approches utilisent des techniques de décomposition. La classification des résultats numériques montre que la méthode exacte (BB) est efficace pour les problèmes de petite taille, que la méthode OA surpasse les autres méthodes pour les instances de taille moyenne et que l’heuristique proposée de branchement est efficace pour les instances de taille importante où les méthodes BB et OA ne sont pas applicables. À cause de la complexité de la structure du problème, les méthodes exactes sont incapables de résoudre les problèmes de taille importante dans un temps raisonnable. C’est pourquoi les méthodes heuristiques sont développées pour faire un compromis entre la précision de la solution et le temps de résolution.----------ABSTRACT : This thesis investigates the Markowitz’ Mean-Variance (MV) portfolio optimization model with cardinality constraint and bounds on variables which is MIQP model and known as an NP-Hard problem. We evaluate the performance of the optimal MV-portfolio generated by Branch-and-Bound (BB) algorithm as an exact method which provides a global optimal solution in comparison with the most effective alternative methods in literature such as Mean Absolute Deviation, Gini Mean Difference and Conditional Value at Risk. These alternative methods make use of different risk measures. In addition, we applied an Outer Approximation (OA) algorithm for the MV problem for the first time as well as proposing a new Heuristic Branching algorithm to deal with the difficulty of the problem for large instances. The later approaches utilize some sort of problem decomposition. With the classification of the numerical results, we showed that the exact method (BB) is efficient for small size problem while for medium size problem the OA outperforms the other methods and the proposed Heuristic Branching algorithm is efficient for large size problems since BB and OA are not applicable in this category. Due to the complexity of the problem structure, exact methods are not capable of solving large size problem in a reasonable time budget. Thus, heuristic methods are developed to trade-off between the precision of the solution and computational time

Advanced Optimization and Statistical Methods in Portfolio Optimization and Supply Chain Management

This dissertation is on advanced mathematical programming with applications in portfolio optimization and supply chain management. Specifically, this research started with modeling and solving large and complex optimization problems with cone constraints and discrete variables, and then expanded to include problems with multiple decision perspectives and nonlinear behavior. The original work and its extensions are motivated by real world business problems.The first contribution of this dissertation, is to algorithmic work for mixed-integer second-order cone programming problems (MISOCPs), which is of new interest to the research community. This dissertation is among the first ones in the field and seeks to develop a robust and effective approach to solving these problems. There is a variety of important application areas of this class of problems ranging from network reliability to data mining, and from finance to operations management.This dissertation also contributes to three applications that require the solution of complex optimization problems. The first two applications arise in portfolio optimization, and the third application is from supply chain management. In our first study, we consider both single- and multi-period portfolio optimization problems based on the Markowitz (1952) mean/variance framework. We have also included transaction costs, conditional value-at-risk (CVaR) constraints, and diversification constraints to approach more realistic scenarios that an investor should take into account when he is constructing his portfolio. Our second work proposes the empirical validation of posing the portfolio selection problem as a Bayesian decision problem dependent on mean, variance and skewness of future returns by comparing it with traditional mean/variance efficient portfolios. The last work seeks supply chain coordination under multi-product batch production and truck shipment scheduling under different shipping policies. These works present a thorough study of the following research foci: modeling and solution of large and complex optimization problems, and their applications in supply chain management and portfolio optimization.Ph.D., Business Administration -- Drexel University, 201

Stochastic programming models and methods for portfolio optimization and risk management

This project is focused on stochastic models and methods and their application in portfolio optimization and risk management. In particular it involves development and analysis of novel numerical methods for solving these types of problem. First, we study new numerical methods for a general second order stochastic dominance model where the underlying functions are not necessarily linear.Specifically, we penalize the second order stochastic dominance constraints to the objective under Slater’s constraint qualification and then apply the well known stochastic approximation method and the level function methods to solve the penalized problem and present the corresponding convergence analysis. All methods are applied to some portfolio optimization problems, where the underlying functions are not necessarily linear all results suggests that the portfolio strategy generated by the second order stochastic dominance model outperform the strategy generated by the Markowitz model in a sense of having higher return and lower risk. Furthermore a nonlinear supply chain problem is considered, where the performance of the level function method is compared to the cutting plane method. The results suggests that the level function method is more efficient in a sense of having lower CPU time as well as being less sensitive to the problem size. This is followed by study of multivariate stochastic dominance constraints. We propose a penalization scheme for the multivariate stochastic dominance constraint and present the analysis regarding the Slater constraint qualification. The penalized problem is solved by the level function methods and a modified cutting plane method and compared to the cutting surface method proposed in [70] and the linearized method proposed in [4]. The convergence analysis regarding the proposed algorithms are presented. The proposed numerical schemes are applied to a generic budget allocation problem where it is shown that the proposed methods outperform the linearized method when the problem size is big. Moreover, a portfolio optimization problem is considered where it is shown that the a portfolio strategy generated by the multivariate second order stochastic dominance model outperform the portfolio strategy generated by the Markowitz model in sense of having higher return and lower risk. Also the performance of the algorithms is investigated with respect to the computation time and the problem size. It is shown that the level function method and the cutting plane method outperform the cutting surface method in a sense of both having lower CPU time as well as being less sensitive to the problem size. Finally, reward-risk analysis is studied as an alternative to stochastic dominance. Specifically, we study robust reward-risk ratio optimization. We propose two robust formulations, one based on mixture distribution, and the other based on the first order moment approach. We propose a sample average approximation formulation as well as a penalty scheme for the two robust formulations respectively and solve the latter with the level function method. The convergence analysis are presented and the proposed models are applied to Sortino ratio and some numerical test results are presented. The numerical results suggests that the robust formulation based on the first order moment results in the most conservative portfolio strategy compared to the mixture distribution model and the nominal model

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

Risk neutral and risk averse stochastic optimization

In this thesis, we focus on the modeling, computational methods and applications of multistage/single-stage stochastic optimization, which entail risk aversion under certain circumstances. Chapters 2-4 concentrate on multistage stochastic programming while Chapter 5-6 deal with a class of single-stage functional constrained stochastic optimization problems. First, we investigate the deterministic upper bound of a Multistage Stochastic Linear Program (MSLP). We first present the Dual SDDP algorithm, which solves the Dynamic Programming equations for the dual and computes a sequence of nonincreasing deterministic upper bounds for the optimal value of the problem, even without the presence of Relatively Complete Recourse (RCR) condition. We show that optimal dual solutions can be obtained using Primal SDDP when computing the duals of the subproblems in the backward pass. As a byproduct, we study the sensitivity of the optimal value as a function of the involved problem parameters. In particular, we provide formulas for the derivatives of the value function with respect to the parameters and illustrate their application on an inventory problem. Next, we extend to the infinite-horizon MSLP and show how to construct a deterministic upper bound (dual bound) via the proposed Periodical Dual SDDP. Finally, as a proof of concept of the developed tools, we present the numerical results of (1) the sensitivity of the optimal value as a function of the demand process parameters; (2) conduct Dual SDDP on the inventory and the Brazilian hydro-thermal planning problems under both finite-horizon and infinite-horizon settings. Third, we discuss sample complexity of solving stationary stochastic programs by the Sample Average Approximation (SAA) method. We investigate this in the framework of Stochastic Optimal Control (in discrete time) setting. In particular we derive a Central Limit Theorem type asymptotics for the optimal values of the SAA problems. The main conclusion is that the sample size, required to attain a given relative error of the SAA solution, is not sensitive to the discount factor, even if the discount factor is very close to one. We consider the risk neutral and risk averse settings. The presented numerical experiments confirm the theoretical analysis. Fourth, we propose a novel projection-free method, referred to as Level Conditional Gradient (LCG) method, for solving convex functional constrained optimization. Different from the constraint-extrapolated conditional gradient type methods (CoexCG and CoexDurCG), LCG, as a primal method, does not assume the existence of an optimal dual solution, thus improving the convergence rate of CoexCG/CoexDurCG by eliminating the dependence on the magnitude of the optimal dual solution. Similar to existing level-set methods, LCG uses an approximate Newton method to solve a root-finding problem. In each approximate Newton update, LCG calls a conditional gradient oracle (CGO) to solve a saddle point subproblem. The CGO developed herein employs easily computable lower and upper bounds on these saddle point problems. We establish the iteration complexity of the CGO for solving a general class of saddle point optimization. Using these results, we show that the overall iteration complexity of the proposed LCG method is $\mathcal{O}\left(\frac{1}{\epsilon^2}\log(\frac{1}{\epsilon})\right)$ for finding an $\epsilon$-optimal and $\epsilon$-feasible solution of the considered problem. To the best of our knowledge, LCG is the first primal conditional gradient method for solving convex functional constrained optimization. For the subsequently developed nonconvex algorithms in this thesis, LCG can also serve as a subroutine or provide high-quality starting points that expedites the solution process. Last, to cope with the nonconvex functional constrained optimization problems, we develop three approaches: the Level Exact Proximal Point (EPP-LCG) method, the Level Inexact Proximal Point (IPP-LCG) method and the Direct Nonconvex Conditional Gradient (DNCG) method. The proposed EPP-LCG and IPP-LCG methods utilize the proximal point framework and solve a series of convex subproblems. By solving each subproblem, they leverage the proposed LCG method, thus averting the effect from large Lagrangian multipliers. We show that the iteration complexity of the algorithms is bounded by $\mathcal{O}\left(\frac{1}{\epsilon^3}\log(\frac{1}{\epsilon})\right)$ in order to obtain an (approximate) KKT point. However, the proximal-point type methods have triple-layer structure and may not be easily implementable. To alleviate the issue, we also propose the DNCG method, which is the first single-loop projection-free algorithm for solving nonconvex functional constrained problem in the literature. This algorithm provides a drastically simpler framework as it only contains three updates in one loop. We show that the iteration complexity to find an $\epsilon$-Wolfe point is bounded by $\mathcal{O}\big(1/{\epsilon^4}\big)$. To the best of our knowledge, all these developments are new for projection-free methods for nonconvex optimization. We demonstrate the effectiveness of the proposed nonconvex projection-free methods on a portfolio selection problem and the intensity modulated radiation therapy treatment planning problem. Moreover, we compare the results with the LCG method proposed in Chapter \ref{chapter-noncvx}. The outcome of the numerical study shows all methods are efficient in jointly minimizing risk while promoting sparsity in a rather short computational time for the real-world and large-scale datasets.Ph.D

Robust asset allocation problems: a new class of risk measure based models

Many optimization problems involve parameters which are not known in advance, but can only be forecast or estimated. This is true, for example, in portfolio asset allocation. Such problems fit perfectly into the framework of Robust Optimization that, given optimization problems with uncertain parameters, looks for solutions that will achieve good objective function values for the realization of these parameters in given uncertainty sets. Aim of this dissertation is to compare alternative forms of robustness in the context of portfolio asset allocation. Starting with the concept of convex risk measures, a new family of models, called "norm-portfolio" models, is firstly proposed where not only the values of the uncertainty parameters, but also their degree of feasibility are specified. This relaxed form of robustness is obtained by exploiting the link between convex risk measures and classical robustness. Then, we test some norm-portfolio models, as well as various robust strategies from the literature, with real market data on three different data sets. The objective of the computational study is to compare alternative forms of relaxed robustness - the relaxed robustness characterizing the "norm-portfolio" models, the so-called soft robustness and the CVaR robustness. In addition, the models above are compared to a more classical robust model from the literature, in order to experiment similarities and dissimilarities between robust models based on convex risk measures and more traditional robust approaches

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more