Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices

Combining Alpha Streams with Costs

We discuss investment allocation to multiple alpha streams traded on the same execution platform with internal crossing of trades and point out differences with allocating investment when alpha streams are traded on separate execution platforms with no crossing. First, in the latter case allocation weights are non-negative, while in the former case they can be negative. Second, the effects of both linear and nonlinear (impact) costs are different in these two cases due to turnover reduction when the trades are crossed. Third, the turnover reduction depends on the universe of traded alpha streams, so if some alpha streams have zero allocations, turnover reduction needs to be recomputed, hence an iterative procedure. We discuss an algorithm for finding allocation weights with crossing and linear costs. We also discuss a simple approximation when nonlinear costs are added, making the allocation problem tractable while still capturing nonlinear portfolio capacity bound effects. We also define "regression with costs" as a limit of optimization with costs, useful in often-occurring cases with singular alpha covariance matrix.Comment: 21 pages; minor misprints corrected; to appear in The Journal of Ris

MIPaaL: Mixed Integer Program as a Layer

Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems from specific classes with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a Mixed Integer Linear Program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, which is an exact algorithm that iteratively adds constraints to a continuous relaxation of the problem until an integral solution is found. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and prescription separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP

Landscape Surrogate: Learning Decision Losses for Mathematical Optimization Under Partial Information

Recent works in learning-integrated optimization have shown promise in settings where the optimization problem is only partially observed or where general-purpose optimizers perform poorly without expert tuning. By learning an optimizer $\mathbf{g}$ to tackle these challenging problems with $f$ as the objective, the optimization process can be substantially accelerated by leveraging past experience. The optimizer can be trained with supervision from known optimal solutions or implicitly by optimizing the compound function $f\circ \mathbf{g}$. The implicit approach may not require optimal solutions as labels and is capable of handling problem uncertainty; however, it is slow to train and deploy due to frequent calls to optimizer $\mathbf{g}$ during both training and testing. The training is further challenged by sparse gradients of $\mathbf{g}$, especially for combinatorial solvers. To address these challenges, we propose using a smooth and learnable Landscape Surrogate $M$ as a replacement for $f\circ \mathbf{g}$. This surrogate, learnable by neural networks, can be computed faster than the solver $\mathbf{g}$, provides dense and smooth gradients during training, can generalize to unseen optimization problems, and is efficiently learned via alternating optimization. We test our approach on both synthetic problems, including shortest path and multidimensional knapsack, and real-world problems such as portfolio optimization, achieving comparable or superior objective values compared to state-of-the-art baselines while reducing the number of calls to $\mathbf{g}$. Notably, our approach outperforms existing methods for computationally expensive high-dimensional problems

A Tool for Visually Exploring Multi-objective Mixed-Integer Optimization Models

Multi-objective optimization models have been increasingly used as optimal decisions are searched in settings considering several conflicting objectives. In these cases compromises must be made and often a large number of nondominated optimal solutions exist. From these solutions decisionmakers must find the preferred one. This is a difficult task both from a computational and cognitive point of views, as it requires several solutions to be obtained and compared. An interactive visualization tool for fully understanding the best trade-offs is therefore becoming increasingly important. This paper proposes visualization solutions, implemented in a tool, for aiding decision-makers in finding the preferred solution in multiobjective optimization problems

Heuristic analysis of investment strategy

The present article investigates the problem of optimal investment, when, given a limited amount of funds, a decision must be taken to which projects and what amounts of funds are to be invested. Supposing that the expected average profit depends on several possible different market conditions, a matrix .game against nature. has been selected as the initial mathematical model. With a view to develop the optimal investment strategy, a linear programming task is formulated. The sensitivity of solutions to profitability coefficients is analysed by means of formulating a dual task for this task. The present article considers the stability and dynamics of the optimum investment strategy given a varying amount of the funds allocated to investment and the profitability of specific projects. Investavimo strategijos euristinė analizė Santrauka Nagrinėjama optimalaus investavimo problema, kai, turint ribotą lėšų kiekį, reikia nuspręsti, į kuriuos projektus ir kokias sumas turėtume investuoti. Teigiant, kad laukiamas vidutinis pelnas priklauso nuo kelių galimų skirtingų rinkos būsenų, pradiniu matematiniu modeliu pasirenkamas matricinis lošimas su gamta. Optimaliai investavimo strategijai gauti formuluojamas tiesinio programavimo uždavinys, kurio optimali tikslo funkcijos reikšmė yra garantuotas, nuo rinkos būsenos nepriklausantis vidutinis pelnas. Naudojant šiam uždaviniui dualųjį uždavinį, tiriamas sprendinio jautrumas pelno koeficientams. Parametrizuojant abiejų, tiesioginio ir dualiojo, tiesinio programavimo uždavinių koeficientus, nagrinėjamas optimalios investavimo strategijos stabilumas ir dinamika, kintant investuotojui skirtų lėšų kiekiui bei atskirų projektų pelningumui. First Published Online: 21 Oct 2010 Reikšminiai žodžiai: optimalus investavimas, matricinis lošimas, parametrinis programavimas, garantuotas vidutinis pelnas

Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices