Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems

We consider nonlinear optimization problems with cardinality constraints. Based on a continuous reformulation, we introduce second-order necessary and sufficient optimality conditions. Under such a second-order condition, we can guarantee local uniqueness of Mordukhovich stationary points. Finally, we use this observation to provide extended local convergence theory for a Scholtes-type regularization method, which guarantees the existence and convergence of iterates under suitable assumptions. This convergence theory can also be applied to other regularization schemes.     

Integer programming approaches in mean-risk models

This paper is concerned with porfolio optimization problems with integer constraints. Such problems include, among others mean-risk problems with nonconvex transaction cost, minimal transaction unit constraints and cardinality constraints on the number of assets in a portfolio. These problems, though practically very important have been considered intractable because we have to solve nonlinear integer programming problems for which there exists no efficient algorithms. We will show that these problems can now be solved by the state- of-the-art integer programming methodologies if we use absolute deviation as the measure of risk.     

Algorithm for cardinality-constrained quadratic optimization

This paper describes an algorithm for cardinality-constrained quadratic optimization problems, which are convex quadratic programming problems with a limit on the number of non-zeros in the optimal solution. In particular, we consider problems of subset selection in regression and portfolio selection in asset management and propose branch-and-bound based algorithms that take advantage of the special structure of these problems. We compare our tailored methods against CPLEX’s quadratic mixed-integer solver and conclude that the proposed algorithms have practical advantages for the special class of problems we consider. The research of D. Bertsimas was partially supported by the Singapore-MIT alliance. The research of R. Shioda was partially supported by the Singapore-MIT alliance, the Discovery Grant from NSERC and a research grant from the Faculty of Mathematics, University of Waterloo.     

Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA

In this paper, we consider the case of downside risk measures with cardinality and bounding constraints in portfolio selection. These constraints limit the amount of capital to be invested in each asset as well as the number of assets composing the portfolio. While the standard Markowitz’s model is a convex quadratic program, this new model is a NP-hard mixed integer quadratic program. Realizing the computational intractability for this class of problems, especially large-scale problems, we first reformulate it as a DC program with the help of exact penalty techniques in Difference of Convex functions (DC) programming and then solve it by DC Algorithms (DCA). To check globality of computed solutions, a global method combining the local algorithm DCA with a Branch-and-Bound algorithm is investigated. Numerical simulations show that DCA is an efficient and promising approach for the considered problem.      

Warm-Start Heuristic for Stochastic Portfolio Optimization with Fixed and Proportional Transaction Costs

We consider a probabilistic portfolio optimization model including fixed and proportional transaction costs. We derive a deterministic equivalent of the probabilistic model for fat-tailed portfolio returns. We develop a method which finds provably near-optimal solutions in minimal amount of time for industry-sized (up to 2000 assets) problems. To solve the mixed-integer nonlinear programming (MINLP) deterministic formulation equivalent to the stochastic problem, we design a mathematical programming-based warm-start heuristic. The tests show the computational efficiency of the heuristic which is more than an order of magnitude faster than Cplex in finding high-quality solutions.     

Solving continuous min max problem for single period portfolio selection with discrete constraints by DCA

In this article, we consider the application of a continuous min–max model with cardinality constraints to worst-case portfolio selection with multiple scenarios of risk, where the return forecast of each asset belongs to an interval. The problem can be formulated as minimizing a convex function under mixed integer variables with additional complementarity constraints. We first prove that the complementarity constraints can be eliminated and then use Difference of Convex functions (DC) programming and DC Algorithm (DCA), an innovative approach in non-convex programming frameworks, to solve the resulting problem. We reformulate it as a DC program and then show how to apply DCA to solve it. Numerical experiments on several test problems are reported that demonstrate the accuracy of the proposed algorithm.     

Long-Short Portfolio Optimization Under Cardinality Constraints by Difference of Convex Functions Algorithm

In the matter of Portfolio selection, we consider an extended version of the Mean-Absolute Deviation (MAD) model, which includes discrete asset choice constraints (threshold and cardinality constraints) and one is allowed to sell assets short if it leads to a better risk-return tradeoff. Cardinality constraints limit the number of assets in the optimal portfolio and threshold constraints limit the amount of capital to be invested in (or sold short from) each asset and prevent very small investments in (or short selling from) any asset. The problem is formulated as a mixed 0–1 programming problem, which is known to be NP-hard. Attempting to use DC (Difference of Convex functions) programming and DCA (DC Algorithms), an efficient approach in non-convex programming framework, we reformulate the problem in terms of a DC program, and investigate a DCA scheme to solve it. Some computational results carried out on benchmark data sets show that DCA has a better performance in comparison to the standard solver IBM CPLEX.     

Solving non-linear portfolio optimization problems with the primal-dual interior point method

Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve NLP problems of typical sizes. However stochastic programming problems are highly structured. The key to the efficient solution of such problems is therefore the ability to exploit their structure. Interior point methods are well-suited to the solution of very large non-linear optimization problems. In this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables, featuring constraints on semi-variance, skewness or non-linear utility functions in the objective, can be solved with the state-of-the-art solver.     

The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming

A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or quadratic integer-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original MINLP problem. After an initial overestimated set has been obtained the algorithm solves a sequence of mixed-integer linear programming or mixed-integer quadratic programming subproblems and refines the overestimated set by generating more supporting hyperplanes in each iteration. Compared to the extended cutting plane algorithm ESH generates a tighter overestimated set and unlike outer approximation the generation point for the supporting hyperplanes is found by a simple line search procedure. In this paper it is proven that the ESH algorithm converges to a global optimum for convex MINLP problems. The ESH algorithm is implemented as the supporting hyperplane optimization toolkit (SHOT) solver, and an extensive numerical comparison of its performance against other state-of-the-art MINLP solvers is presented.     

Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach

In this paper we consider cardinality-constrained convex programs that minimize a convex function subject to a cardinality constraint and other linear constraints. This class of problems has found many applications, including portfolio selection, subset selection and compressed sensing. We propose a successive convex approximation method for this class of problems in which the cardinality function is first approximated by a piecewise linear DC function (difference of two convex functions) and a sequence of convex subproblems is then constructed by successively linearizing the concave terms of the DC function. Under some mild assumptions, we establish that any accumulation point of the sequence generated by the method is a KKT point of the DC approximation problem. We show that the basic algorithm can be refined by adding strengthening cuts in the subproblems. Finally, we report some preliminary computational results on cardinality-constrained portfolio selection problems.     

Minimal Zero Norm Solutions of Linear Complementarity Problems

In this paper, we study minimal zero norm solutions of the linear complementarity problems, defined as the solutions with smallest cardinality. Minimal zero norm solutions are often desired in some real applications such as bimatrix game and portfolio selection. We first show the uniqueness of the minimal zero norm solution for Z-matrix linear complementarity problems. To find minimal zero norm solutions is equivalent to solve a difficult zero norm minimization problem with linear complementarity constraints. We then propose a p norm regularized minimization model with p in the open interval from zero to one, and show that it can approximate minimal zero norm solutions very well by sequentially decreasing the regularization parameter. We establish a threshold lower bound for any nonzero entry in its local minimizers, that can be used to identify zero entries precisely in computed solutions. We also consider the choice of regularization parameter to get desired sparsity. Based on the theoretical results, we design a sequential smoothing gradient method to solve the model. Numerical results demonstrate that the sequential smoothing gradient method can effectively solve the regularized model and get minimal zero norm solutions of linear complementarity problems.     

Conic mixed-integer rounding cuts

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean–variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability. This research has been supported, in part, by Grant # DMI0700203 from the National Science Foundation.     

Efficient projected gradient methods for cardinality constrained optimization

Sparse optimization has attracted increasing attention in numerous areas such as compressed sensing, financial optimization and image processing. In this paper, we first consider a special class of cardinality constrained optimization problems, which involves box constraints and a singly linear constraint. An efficient approach is provided for calculating the projection over the feasibility set after a careful analysis on the projection subproblem. Then we present several types of projected gradient methods for a general class of cardinality constrained optimization problems. Global convergence of the methods is established under suitable assumptions. Finally, we illustrate some applications of the proposed methods for signal recovery and index tracking.Especially for index tracking, we propose a new model subject to an adaptive upper bound on the sparse portfolio weights. The computational results demonstrate that the proposed projected gradient methods are efficient in terms of solution quality.     

On portfolio optimization using fuzzy decisions

We consider stochastic optimization problems involving stochastic dominance constraints. We develop portfolio optimization model involving stochastic dominance constrains using fuzzy decisions and we concentrate on fuzzy linear programming problems with only fuzzy technological coefficients and aplication/implementation of modified subgradient method to fuzy linear programming problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Using medians in portfolio optimization

Some new portfolio optimization models are formulated by adopting the sample median instead of the sample mean as the investment efficiency measure. The median is a robust statistic, which is less affected by outliers than the mean, and in portfolio models this is particularly relevant as data are often characterized by attributes such as skewness, fat tails and jumps, which may strongly bias the mean estimate. As in mean/variance optimization, the portfolio problems are formulated as finding the optimal weights, for example, wealth allocation, which maximize the portfolio median, with risk constrained by some risk measure, respectively, the Value-at-Risk, the Conditional Value-at-Risk, the Mean Absolute Deviation and the Maximum Loss, for a whole of four different models. All these models are formulated as mixed integer linear programming problems, which, at least for moderate sized problems, are efficiently solved by standard software. Models are tested on real financial data, compared to some benchmark portfolios, and found to give good results in terms of realized profits. An important feature is greater portfolio diversification than that obtained with other portfolio models.     

Coradiant sets and $$\varepsilon $$-efficiency in multiobjective optimization

In multi-parametric programming an optimization problem is solved as a function of certain parameters, where the parameters are commonly considered to be bounded and continuous. In this paper, we use the case of strictly convex multi-parametric quadratic programming (mp-QP) problems with affine constraints to investigate problems where these conditions are not met. Based on the combinatorial solution approach for mp-QP problems featuring bounded and continuous parameters, we show that (i) for unbounded parameters, it is possible to obtain the multi-parametric solution if there exists one realization of the parameters for which the optimization problem can be solved and (ii) for binary parameters, we present the equivalent mixed-integer formulations for the application of the combinatorial algorithm. These advances are combined into a new, generalized version of the combinatorial algorithm for mp-QP problems, which enables the solution of problems featuring both unbounded and binary parameters. This novel approach is applied to mixed-integer bilevel optimization problems and the parametric solution of the dual of a convex problem.     

A local relaxation method for the cardinality constrained portfolio optimization problem

The NP-hard nature of cardinality constrained mean-variance portfolio optimization problems has led to a number of different algorithms with varying degrees of success in reaching optimality given limited computational resources and under the presence of strict time constraints present in practice. The proposed local relaxation algorithm explores the inherent structure of the objective function. It solves a sequence of small, local, quadratic-programs by first projecting asset returns onto a reduced metric space, followed by clustering in this space to identify sub-groups of assets that best accentuate a suitable measure of similarity amongst different assets. The algorithm can either be cold started using a suitable heuristic method such as the centroids of initial clusters or be warm started based on the last output. Results, using a basket of up to 3,000 stocks and with different cardinality constraints, indicates that the proposed algorithm can lead to significant performance gain over popular branch-and-cut methods. One key application of this algorithm is in dealing with large scale cardinality constrained portfolio optimization under tight time constraint, such as for the purpose of index tracking or index arbitrage at high frequency.     

Nested Conditional Value-at-Risk portfolio selection: A model with temporal dependence driven by market-index volatility

In a multistage stochastic programming framework, we develop a new method for finding an approximated portfolio allocation solution to the nested Conditional Value-at-Risk model when asset log returns are stagewise dependent. We describe asset log returns through a single-factor model where the driving factor is the market-index log return modeled by a Generalized Autoregressive Conditional Heteroskedasticity process to take into account the serial dependence usually observed. To solve the nested Conditional Value-at-Risk model, we implement a backward induction scheme coupled with cubic spline interpolation that reduces the computational complexity of the optimal portfolio allocation and allows to treat problems otherwise unmanageable.     

Portfolio optimization with an envelope-based multi-objective evolutionary algorithm

The problem of portfolio selection is a standard problem in financial engineering and has received a lot of attention in recent decades. Classical mean–variance portfolio selection aims at simultaneously maximizing the expected return of the portfolio and minimizing portfolio variance. In the case of linear constraints, the problem can be solved efficiently by parametric quadratic programming (i.e., variants of Markowitz’ critical line algorithm). However, there are many real-world constraints that lead to a non-convex search space, e.g., cardinality constraints which limit the number of different assets in a portfolio, or minimum buy-in thresholds. As a consequence, the efficient approaches for the convex problem can no longer be applied, and new solutions are needed.In this paper, we propose to integrate an active set algorithm optimized for portfolio selection into a multi-objective evolutionary algorithm (MOEA). The idea is to let the MOEA come up with some convex subsets of the set of all feasible portfolios, solve a critical line algorithm for each subset, and then merge the partial solutions to form the solution of the original non-convex problem. We show that the resulting envelope-based MOEA significantly outperforms existing MOEAs.     

Global optimization of mixed-integer bilevel programming problems

Two approaches that solve the mixed-integer nonlinear bilevel programming problem to global optimality are introduced. The first addresses problems mixed-integer nonlinear in outer variables and C²-nonlinear in inner variables. The second adresses problems with general mixed-integer nonlinear functions in outer level. Inner level functions may be mixed-integer nonlinear in outer variables, linear, polynomial, or multilinear in inner integer variables, and linear in inner continuous variables. This second approach is based on reformulating the mixed-integer inner problem as continuous via its vertex polyheral convex hull representation and solving the resulting nonlinear bilevel optimization problem by a novel deterministic global optimization framework. Computational studies illustrate proposed approaches.