The cardinality-constrained mean-variance portfolio problem has garnered significant attention within contemporary finance due to its potential for achieving low risk while effectively managing risks and transaction costs. Instead of solving this problem directly, many existing methods rely on regularization and approximation techniques, which hinder investors' ability to precisely specify a portfolio's desired cardinality level. Moreover, these approaches typically include more hyper-parameters and increase the problem's dimensionality. To address these challenges, we propose a customized penalty decomposition algorithm. We demonstrate that this algorithm not only does it converge to a local minimizer of the cardinality-constrained mean-variance portfolio problem, but is also computationally efficient. Our approach leverages a sequence of penalty subproblems, each tackled using Block Coordinate Descent (BCD). Specifically, we show that the steps within BCD yield closed-form solutions, allowing us to identify a saddle point of the penalty subproblems. Finally, by applying our penalty decomposition algorithm to real-world datasets, we highlight its efficiency and its superiority over state-of-the-art methods across several performance metrics.
We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption o…
This paper studies mean-risk portfolio optimization models using the conditional value-at-risk (CVaR) as a risk measure. We also employ a cardinality constraint for limiting the number of invested assets. Solving such a cardinality-constrained mean-CVaR model is computationally challenging for two main reasons. First, this model is formulated as a mixed-integer optimization (MIO) problem because of the cardinality constraint, so solving it exactly is very hard when the number of investable assets is large. Second, the problem size depends on the number of asset return scenarios, and the comput…
A cardinality-constrained portfolio caps the number of stocks to be traded across and within groups or sectors. These limitations arise from real-world scenarios faced by fund managers, who are constrained by transaction costs and client preferences as they seek to maximize return and limit risk. We develop a new approach to solve cardinality-constrained portfolio optimization problems, extending both Markowitz and conditional value at risk (CVaR) optimization models with cardinality constraints. We derive a continuous relaxation method for the NP-hard objective, which allows for very efficien…
Mean-reverting portfolios with few assets, but high variance, are of great interest for investors in financial markets. Such portfolios are straightforwardly profitable because they include a small number of assets whose prices not only oscillate predictably around a long-term mean but also possess enough volatility. Roughly speaking, sparsity minimizes trading costs, volatility provides arbitrage opportunities, and mean-reversion property equips investors with ideal investment strategies. Finding such favorable portfolios can be formulated as a nonconvex quadratic optimization problem with an…
The Markowitz-based portfolio selection turns to an NP-hard problem when considering cardinality constraints. In this case, existing exact solutions like quadratic programming may not be efficient to solve the problem. Many researchers, therefore, used heuristic and metaheuristic approaches in order to deal with the problem. This work presents Asexual Reproduction Optimization (ARO), a model free metaheuristic algorithm inspired by the asexual reproduction, in order to solve the portfolio optimization problem including cardinality constraint to ensure the investment in a given number of differ…
This paper studies a distributionally robust portfolio optimization model with a cardinality constraint for limiting the number of invested assets. We formulate this model as a mixed-integer semidefinite optimization (MISDO) problem by means of the moment-based ambiguity set of probability distributions of asset returns. To exactly solve large-scale problems, we propose a specialized cutting-plane algorithm that is based on bilevel optimization reformulation. We prove the finite convergence of the algorithm. We also apply a matrix completion technique to lower-level SDO problems to make their …
In this paper, we aim at solving the cardinality constrained high-order portfolio optimization, i.e., mean-variance-skewness-kurtosis model with cardinality constraint (MVSKC). Optimization for the MVSKC model is of great difficulty in two parts. One is that the objective function is non-convex, the other is the combinational nature of the cardinality constraint, leading to non-convexity as well dis-continuity. Based on the observation that cardinality constraint has the difference-of-convex (DC) property, we transform the cardinality constraint into a penalty term and then propose three algor…
In this work, we propose a hybrid variant of the level-based learning swarm optimizer (LLSO) for solving large-scale portfolio optimization problems. Our goal is to maximize a modified formulation of the Sharpe ratio subject to cardinality, box and budget constraints. The algorithm involves a projection operator to deal with these three constraints simultaneously and we implicitly control transaction costs thanks to a rebalancing constraint. We also introduce a suitable exact penalty function to manage the turnover constraint. In addition, we develop an ad hoc mutation operator to modify candi…
We consider convex constrained optimization problems that also include a cardinality constraint. In general, optimization problems with cardinality constraints are difficult mathematical programs which are usually solved by global techniques from discrete optimization. We assume that the region defined by the convex constraints can be written as the intersection of a finite collection of convex sets, such that it is easy and inexpensive to project onto each one of them (e.g., boxes, hyper-planes, or half-spaces). Taking advantage of a recently developed continuous reformulation that relaxes th…
In this paper, we consider a class of sparse regression problems, whose objective function is the summation of a convex loss function and a cardinality penalty. By constructing a smoothing function for the cardinality function, we propose a projected neural network and design a correction method for solving this problem. The solution of the proposed neural network is unique, global existent, bounded and globally Lipschitz continuous. Besides, we prove that all accumulation points of the proposed neural network have a common support set and a unified lower bound for the nonzero entries. Combini…
We study a cardinality-constrained optimization problem with nonnegative variables in this paper. This problem is often encountered in practice. Firstly we study some properties on the optimal solutions of this optimization problem under some conditions. An equivalent reformulation of the problem under consideration is proposed. Based on the reformulation, we present a successive convex approximation method for the cardinality constrained optimization problem. We prove that the method converges to a KKT point of the reformulation problem. Under some conditions, the KKT points of the reformulat…
In this paper, a continuous and non-convex promoting sparsity fraction function is studied in two sparse portfolio selection models with and without short-selling constraints. Firstly, we study the properties of the optimal solution to the problem $(FP_{a,λ,η})$ including the first-order and the second optimality condition and the lower and upper bound of the absolute value for its nonzero entries. Secondly, we develop the thresholding representation theory of the problem $(FP_{a,λ,η})$. Based on it, we prove the existence of the resolvent operator of gradient of $P_{a}(x)$, calculate its anal…
The constrained $\ell_0$ regularization plays an important role in sparse reconstruction. A widely used approach for solving this problem is the penalty method, of which the least square penalty problem is a special case. However, the connections between global minimizers of the constrained $\ell_0$ problem and its penalty problem have never been studied in a systematic way. This work provides a comprehensive investigation on optimal solutions of these two problems and their connections. We give detailed descriptions of optimal solutions of the two problems, including existence, stability with…
Mean-variance optimization (MVO) is known to be sensitive to estimation error in its inputs. Norm penalization of MVO programs is a regularization technique that can mitigate the adverse effects of estimation error. We augment the standard MVO program with a convex combination of parameterized $L_1$ and $L_2$-norm penalty functions. The resulting program is a parameterized quadratic program (QP) whose dual is a box-constrained QP. We make use of recent advances in neural network architecture for differentiable QPs and present a data-driven framework for optimizing parameterized norm-penalties …
We introduce a solution scheme for portfolio optimization problems with cardinality constraints. Typical portfolio optimization problems are extensions of the classical Markowitz mean-variance portfolio optimization model. We solve such type of problems using a method similar to column generation. In this scheme, the original problem is restricted to a subset of the assets resulting in a master convex quadratic problem. Then the dual information of the master problem is used in a sub-problem to propose more assets to consider. We also consider other extensions to the Markowitz model to diversi…
In this paper, we consider a risk-based optimal investment problem of an insurer in a regime-switching jump diffusion model with noisy memory. Using the model uncertainty modeling, we formulate the investment problem as a zero-sum, stochastic differential delay game between the insurer and the market, with a convex risk measure of the terminal surplus and the Brownian delay surplus over a period $[T-\varrho,T]$. Then, by the BSDE approach, the game problem is solved. Finally, we derive analytical solutions of the game problem, for a particular case of a quadratic penalty function and a numeric…
In this work we study a finite horizon optimal liquidation problem with multiplicative price impact in algorithmic trading, using market orders. We analyze the case when an agent is trading on a market with two financial assets, whose difference of log-prices is modelled with a mean-reverting process. The agent's task is to liquidate an initial position of shares of one of the two financial assets, without having the possibility of trading the other stock. The criterion to be optimized consists in maximising the expected final value of the agent, with a running inventory penalty. The main resu…
This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints are nonconvex and complicated, like cardinality constraints, disjunctive programs, or matrix problems involving rank constraints. By a variable duplication and decomposition strategy, the method presented here explicitly handles these difficult constraints, thus generating iterates which are feasible with respect to them, while the remaining (standard and su…
We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse application fields such as signal and image processing, portfolio selection, or machine learning. The paper discusses general-purpose modeling techniques and broadly applicable as well as problem-specific exact and heuristic solution approaches. While our perspective is that of mathematical optimization, a main goal of this work is to reach out to and build bri…
Hydro storage system optimization is becoming one of the most challenging tasks in Energy Finance. While currently the state-of-the-art of the commercial software in the industry implements mainly linear models, we would like to introduce risk aversion and a generic utility function. At the same time, we aim to develop and implement a computational efficient algorithm, which is not affected by the curse of dimensionality and does not utilize subjective heuristics to prevent it. For the short term power market we propose a simultaneous solution for both dispatch and bidding problems. Following …
A Scalable Gradient-Based Optimization Framework for Sparse Minimum-Variance Portfolio Selection
School of Mathematics and Statistics（数学与统计学学院）；University of New South Wales（新南威尔士大学）；School of Mathematical and Physical Sciences（数学与物理科学学院）；University of Technology Sydney（技术大学悉尼分校）；Bayes Business School（贝叶斯商学院）；City St George’s, University of London（伦敦大学城圣乔治学院）；Faculty of Science and Engineering（科学与工程学院）；Macquarie University（麦考瑞大学）
We investigate a class of constrained sparse regression problem with cardinality penalty, where the feasible set is defined by box constraint, and the loss function is convex, but not necessarily smooth. First, we put forward a smoothing fast iterative hard thresholding (SFIHT) algorithm for solving such optimization problems, which combines smoothing approximations, extrapolation techniques and iterative hard thresholding methods. The extrapolation coefficients can be chosen to satisfy $\sup_k β_k=1$ in the proposed algorithm. We discuss the convergence behavior of the algorithm with differen…
We expand the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka-Łojasiewicz (K-Ł) property holds, this is strengthened to convergence to a single constrained stationary point. Our analysis applies under assumptions that we have endeavored to make as weak as possible. It applies to problems that inv…
University of Pittsburgh（匹兹堡大学）；University of Rochester（罗切斯特大学）
Stochastic composition optimization draws much attention recently and has been successful in many emerging applications of machine learning, statistical analysis, and reinforcement learning. In this paper, we focus on the composition problem with nonsmooth regularization penalty. Previous works either have slow convergence rate or do not provide complete convergence analysis for the general problem. In this paper, we tackle these two issues by proposing a new stochastic composition optimization method for composition problem with nonsmooth regularization penalty. In our method, we apply varian…
Quantum devices can be used to solve constrained combinatorial optimization (COPT) problems thanks to the use of penalization methods to embed the COPT problem's constraints in its objective to obtain a quadratic unconstrained binary optimization (QUBO) reformulation of the COPT. However, the particular way in which this penalization is carried out, affects the value of the penalty parameters, as well as the number of additional binary variables that are needed to obtain the desired QUBO reformulation. In turn, these factors substantially affect the ability of quantum computers to efficiently …
