We recommend an implementation of the Markowitz problem to generate stable portfolios with respect to perturbations of the problem parameters. The stability is obtained proposing novel calibrations of the covariance m...
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We recommend an implementation of the Markowitz problem to generate stable portfolios with respect to perturbations of the problem parameters. The stability is obtained proposing novel calibrations of the covariance matrix between the returns that can be cast as convex or quasiconvex optimization problems. A statistical study as well as a sensitivity analysis of the Markowitz problem allow us to justify these calibrations. Our approach can be used to do a global and explicit sensitivity analysis of a class of quadratic optimization problems. Numerical simulations finally show the benefits of the proposed calibrations using real data.
Cooperative transmission in relay networks is considered, in which a source transmits to its destination with the help of a set of cooperating nodes. The source first transmits locally. The cooperating nodes that rece...
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Cooperative transmission in relay networks is considered, in which a source transmits to its destination with the help of a set of cooperating nodes. The source first transmits locally. The cooperating nodes that receive the source signal retransmit a weighted version of it in an amplify-and-forward (AF) fashion. Assuming knowledge of the second-order statistics of the channel state information, beamforming weights are determined so that the signal-to-noise ratio (SNR) at the destination is maximized subject to two different power constraints, i.e., a total (source and relay) power constraint, and individual relay power constraints. For the former constraint, the original problem is transformed into a problem of one variable, which can be solved via Newton's method. For the latter constraint, this problem is solved completely. It is shown that the semidefinite programming (SDP) relaxation of the original problem always has a rank one solution, and hence the original problem is equivalent to finding the rank one solution of the SDP problem. An explicit construction of such a rank one solution is also provided. Numerical results are presented to illustrate the proposed theoretical findings.
We consider the problem of simultaneous power and code-channel allocation for a secondary transmitter/receiver pair coexisting with a primary code-division multiple-access (CDMA) system. Our objective is to find the o...
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We consider the problem of simultaneous power and code-channel allocation for a secondary transmitter/receiver pair coexisting with a primary code-division multiple-access (CDMA) system. Our objective is to find the optimum transmitting power and code sequence of the secondary channel that maximize the signal-to-interference-plus-noise ratio (SINR) at the output of the maximum SINR linear receiver, while at the same time the SINR of all primary channels at the output of their max-SINR receiver is maintained above a certain threshold. This is a non-convex NP-hard optimization problem. We propose a novel feasible suboptimum solution using semidefinite programming. Simulation studies illustrate the theoretical developments.
In this paper we consider low-rank semidefinite programming (LRSDP) relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem. Using the Gramian representation of a positive semidefinit...
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In this paper we consider low-rank semidefinite programming (LRSDP) relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem. Using the Gramian representation of a positive semidefinite matrix, the LRSDP problem can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function with quadratic equality constraints. For the solution of this problem we propose a continuously differentiable exact merit function that exploits the special structure of the constraints and we use this function to define an efficient and globally convergent algorithm. Finally, we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes.
Given a data matrix, we find its nearest symmetric positive-semidefinite Toeplitz matrix. In this paper, we formulate the problem as an optimization problem with a quadratic objective function and semidefinite constra...
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Given a data matrix, we find its nearest symmetric positive-semidefinite Toeplitz matrix. In this paper, we formulate the problem as an optimization problem with a quadratic objective function and semidefinite constraints. In particular, instead of solving the so-called normal equations, our algorithm eliminates the linear feasibility equations from the start to maintain exact primal and dual feasibility during the course of the algorithm. Subsequently, the search direction is found using an inexact Gauss-Newton method rather than a Newton method on a symmetrized system and is computed using a diagonal preconditioned conjugate-gradient-type method. Computational results illustrate the robustness of the algorithm.
A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads...
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A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and we find equations for the variety of Cayley octads. Interwoven is an exposition of much of the 19th century theory of plane quartics. (C) 2011 Elsevier Ltd. All rights reserved.
A wide variety of problems in global optimization, combinatorial optimization, as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating po...
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A wide variety of problems in global optimization, combinatorial optimization, as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver. It turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver. Numerical results are presented including problems from the SDPLIB and the NETLIB LP library;these libraries contain many ill-conditioned and real-life problems.
We introduce two-stage stochastic semidefinite programs with recourse and present an interior point algorithm for solving these problems using Bender's decomposition. This decomposition algorithm and its analysis ...
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We introduce two-stage stochastic semidefinite programs with recourse and present an interior point algorithm for solving these problems using Bender's decomposition. This decomposition algorithm and its analysis extend Zhao's results [Math. Program., 90 (2001), pp. 507-536] for stochastic linear programs. The convergence results are proved by showing that the logarithmic barrier associated with the recourse function of two-stage stochastic semidefinite programs with recourse is a strongly self-concordant barrier on the first stage solutions. The short-step variant of the algorithm requires O(root p + Kr In mu(0) / epsilon) Newton iterations to follow the first stage central path from a starting value of the barrier parameter mu(0) to a terminating value epsilon. The long-step variant requires O(( p + Kr In mu(0) / epsilon) damped Newton iterations. The calculation of the gradient and Hessian of the recourse function and the first stage Newton direction decomposes across the second stage scenarios.
We present an optimization approach to the weak approximation of a general class of stochastic differential equations with jumps, in particular, when value functions with compact support are considered. Our approach e...
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We present an optimization approach to the weak approximation of a general class of stochastic differential equations with jumps, in particular, when value functions with compact support are considered. Our approach employs a mathematical programming technique yielding upper and lower bounds of the expectation, without Monte Carlo sample paths simulations, based upon the exponential tempering of bounding polynomial functions to avoid their explosion at infinity. The resulting tempered polynomial optimization problems can be transformed into a solvable polynomial programming after a minor approximation. The exponential tempering widens the class of stochastic differential equations for which our methodology is well defined. The analysis is supported by numerical results on the tail probability of a stable subordinator and the survival probability of Ornstein-Uhlenbeck processes driven by a stable subordinator, both of which can be formulated with value functions with compact support and are not applicable in our framework without exponential tempering.
This technical note considers the problem of reducing the computational complexity associated with the Sum-of-Squares approach to stability analysis of time-delay systems. Specifically, this technical note considers s...
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This technical note considers the problem of reducing the computational complexity associated with the Sum-of-Squares approach to stability analysis of time-delay systems. Specifically, this technical note considers systems with a large state-space but where delays affect only certain parts of the system. This yields a coefficient matrix of the delayed state with low rank-a common scenario in practice. The technical note uses the general framework of coupled differential-difference equations with delays in feedback channels. This framework includes systems of both the neutral and retarded-type. The approach is based on recent results which introduced a new Lyapunov-Krasovskii structure which was shown to be necessary and sufficient for stability of this class of systems. This technical note shows how exploiting the structure of the new functional can yield dramatic improvements in computational complexity. Numerical examples are given to illustrate this improvement.
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