A concave optimization-based approach for sparse portfolio selection

Lorenzo, David Di; Liuzzi, Giampaolo; Rinaldi, Francesco; Schoen, Fabio; Sciandrone, Marco

doi:10.1080/10556788.2011.577773

Cited by 32 publications

(21 citation statements)

References 27 publications

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“…This is a classical portfolio optimization problem where Q and μ are the covariance matrix and mean of n possible assets, respectively, and e T x ≤ 1 is a resource constraint; see, e.g., [6,10]. To create test examples, we take the same randomly generated data Q, μ, ρ, and u which were used by Frangioni and Gentile in [13] and which are available at their Webpage [14].…”

Section: Numerical Resultsmentioning

confidence: 99%

Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-Type Conditions and a Regularization Method

Burdakov¹,

Kanzow²,

Schwartz³

2016

SIAM J. Optim.

105

195

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Optimization problems with cardinality constraints are very difficult mathematical programs which are typically solved by global techniques from discrete optimization. Here we introduce a mixed-integer formulation whose standard relaxation still has the same solutions (in the sense of global minima) as the underlying cardinality-constrained problem; the relation between the local minima is also discussed in detail. Since our reformulation is a minimization problem in continuous variables, it allows to apply ideas from that field to cardinality-constrained problems. Here, in particular, we therefore also derive suitable stationarity conditions and suggest an appropriate regularization method for the solution of optimization problems with cardinality constraints. This regularization method is shown to be globally convergent to a Mordukhovich-stationary point. Extensive numerical results are given to illustrate the behavior of this method

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Section: Numerical Resultsmentioning

confidence: 99%

Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-Type Conditions and a Regularization Method

Burdakov¹,

Kanzow²,

Schwartz³

2016

SIAM J. Optim.

105

195

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“…The local minimizers x 1 and x 2 were each found in approximately 6 % of the runs. For the remaining 8 % of the initial points, SNOPT mostly ended up in x 5 . Then, we applied the same method to the problem from Example 2, now choosing x 0 randomly from OE0; 2 3 .…”

Section: Some Numerical Illustrationsmentioning

confidence: 99%

On a Reformulation of Mathematical Programs with Cardinality Constraints

Burdakov

Kanzow

Schwartz

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Mathematical programs with cardinality constraints are optimization problems with an additional constraint which requires the solution to be sparse in the sense that the number of nonzero elements, i.e. the cardinality, is bounded by a given constant. Such programs can be reformulated as a mixed-integer ones in which the sparsity is modeled with the use of complementarity-type constraints. It is shown that the standard relaxation of the integrality leads to a nonlinear optimization program of the striking property that its solutions (global minimizers) are the same as the solutions of the original program with cardinality constraints. Since the number of local minimizers of the relaxed program is typically larger than the number of local minimizers of the cardinality-constrained problem, the relationship between the local minimizers is also discussed in detail. Furthermore, we show under which assumptions the standard KKT conditions are necessary optimality conditions for the relaxed program. The main result obtained for such conditions is significantly different from the existing optimality conditions that are known for the somewhat related class of mathematical programs with complementarity constraints.

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“…Following [17], we consider a classical portfolio optimization problem min x∈R n x T Qx s.t. μ T x ≥ ρ, e T x ≤ 1, 0 ≤ x ≤ u, x 0 ≤ s, (5.1) where Q and μ are the covariance matrix and the mean of n possible assets and e T x ≤ 1 is the budget constraint, see [12,20]. We generated the test problems using the data from [24], considering s = 5, 10, 20 for each dimension n = 200, 300, 400, which resulted in 270 test problems, see also [17].…”

Section: Portfolio Optimization Problemsmentioning

confidence: 99%

An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

Kanzow

Raharja

Schwartz

2021

J Optim Theory Appl

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A reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

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A concave optimization-based approach for sparse portfolio selection

Cited by 32 publications

References 27 publications

Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-Type Conditions and a Regularization Method

Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-Type Conditions and a Regularization Method

On a Reformulation of Mathematical Programs with Cardinality Constraints

An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

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