928 research outputs found

    A Scalable Algorithm For Sparse Portfolio Selection

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    The sparse portfolio selection problem is one of the most famous and frequently-studied problems in the optimization and financial economics literatures. In a universe of risky assets, the goal is to construct a portfolio with maximal expected return and minimum variance, subject to an upper bound on the number of positions, linear inequalities and minimum investment constraints. Existing certifiably optimal approaches to this problem do not converge within a practical amount of time at real world problem sizes with more than 400 securities. In this paper, we propose a more scalable approach. By imposing a ridge regularization term, we reformulate the problem as a convex binary optimization problem, which is solvable via an efficient outer-approximation procedure. We propose various techniques for improving the performance of the procedure, including a heuristic which supplies high-quality warm-starts, a preprocessing technique for decreasing the gap at the root node, and an analytic technique for strengthening our cuts. We also study the problem's Boolean relaxation, establish that it is second-order-cone representable, and supply a sufficient condition for its tightness. In numerical experiments, we establish that the outer-approximation procedure gives rise to dramatic speedups for sparse portfolio selection problems.Comment: Submitted to INFORMS Journal on Computin

    Bad semidefinite programs: they all look the same

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    Conic linear programs, among them semidefinite programs, often behave pathologically: the optimal values of the primal and dual programs may differ, and may not be attained. We present a novel analysis of these pathological behaviors. We call a conic linear system Ax<=bAx <= b {\em badly behaved} if the value of sup⁑∣Ax<=b\sup { | A x <= b } is finite but the dual program has no solution with the same value for {\em some} c.c. We describe simple and intuitive geometric characterizations of badly behaved conic linear systems. Our main motivation is the striking similarity of badly behaved semidefinite systems in the literature; we characterize such systems by certain {\em excluded matrices}, which are easy to spot in all published examples. We show how to transform semidefinite systems into a canonical form, which allows us to easily verify whether they are badly behaved. We prove several other structural results about badly behaved semidefinite systems; for example, we show that they are in NP∩coβˆ’NPNP \cap co-NP in the real number model of computing. As a byproduct, we prove that all linear maps that act on symmetric matrices can be brought into a canonical form; this canonical form allows us to easily check whether the image of the semidefinite cone under the given linear map is closed.Comment: For some reason, the intended changes between versions 4 and 5 did not take effect, so versions 4 and 5 are the same. So version 6 is the final version. The only difference between version 4 and version 6 is that 2 typos were fixed: in the last displayed formula on page 6, "7" was replaced by "1"; and in the 4th displayed formula on page 12 "A_1 - A_2 - A_3" was replaced by "A_3 - A_2 - A_1

    A Global Approach for Solving Edge-Matching Puzzles

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    We consider apictorial edge-matching puzzles, in which the goal is to arrange a collection of puzzle pieces with colored edges so that the colors match along the edges of adjacent pieces. We devise an algebraic representation for this problem and provide conditions under which it exactly characterizes a puzzle. Using the new representation, we recast the combinatorial, discrete problem of solving puzzles as a global, polynomial system of equations with continuous variables. We further propose new algorithms for generating approximate solutions to the continuous problem by solving a sequence of convex relaxations

    Low rank matrix recovery from rank one measurements

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    We study the recovery of Hermitian low rank matrices X∈CnΓ—nX \in \mathbb{C}^{n \times n} from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form ajajβˆ—a_j a_j^* for some measurement vectors a1,...,ama_1,...,a_m, i.e., the measurements are given by yj=tr(Xajajβˆ—)y_j = \mathrm{tr}(X a_j a_j^*). The case where the matrix X=xxβˆ—X=x x^* to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements, yj=∣⟨x,aj⟩∣2y_j = |\langle x,a_j\rangle|^2 via the PhaseLift approach, which has been introduced recently. We derive bounds for the number mm of measurements that guarantee successful uniform recovery of Hermitian rank rr matrices, either for the vectors aja_j, j=1,...,mj=1,...,m, being chosen independently at random according to a standard Gaussian distribution, or aja_j being sampled independently from an (approximate) complex projective tt-design with t=4t=4. In the Gaussian case, we require mβ‰₯Crnm \geq C r n measurements, while in the case of 44-designs we need mβ‰₯Crnlog⁑(n)m \geq Cr n \log(n). Our results are uniform in the sense that one random choice of the measurement vectors aja_j guarantees recovery of all rank rr-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 44-designs generalizes and improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.Comment: 24 page
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