We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of convex algebraic geometry. More precisely, we determine in which dimen...
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We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of convex algebraic geometry. More precisely, we determine in which dimensions n this convex semialgebraic set is a spectrahedron or a spectrahedral shadow;in particular, for n >= 5, these give new counterexamples to the Helton-Nie conjecture. Moreover, efficient algorithms are provided if n = 4 to test membership in such a set. For n >= 5, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.
One of the current challenges of risk modelling consists in building global risk models from local ones: from a set of local market risk forecasts (local covariance matrices) and cross-market correlations, a global co...
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One of the current challenges of risk modelling consists in building global risk models from local ones: from a set of local market risk forecasts (local covariance matrices) and cross-market correlations, a global covariance matrix preserving local market estimations and restoring a positive semidefinite matrix must be computed. Convex optimisation, taking advantage of the convex properties of dual functions, is an original and high-performing approach for such a process. In this paper, a particular semidefinite program is posed and solved with dual convex algorithms for correlation matrices in order to build a global risk model, starting from a set local market covariance, and cross-correlation. Some numerical illustrations are given.
A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice a...
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A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice and computationally more effective solvers are needed. This paper investigates how to exploit structure and problem knowledge in some control applications. It is shown that inexact search directions are useful to reduce the computational burden and that operator formalism can be utilized to derive tailored calculations.
This paper proposes a new semidefinite programming relaxation for the satisfiability problem. This relaxation is an extension of previous relaxations arising from the paradigm of partial semidefinite liftings for 0/1 ...
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This paper proposes a new semidefinite programming relaxation for the satisfiability problem. This relaxation is an extension of previous relaxations arising from the paradigm of partial semidefinite liftings for 0/1 optimization problems. The construction of the relaxation depends on a choice of permutations of the clauses, and different choices may lead to different relaxations. We then consider the Tseitin instances, a class of instances known to be hard for certain proof systems, and prove that for any choice of permutations, the proposed relaxation is exact for these instances, meaning that a Tseitin instance is unsatisfiable if and only if the corresponding semidefinite programming relaxation is infeasible.
semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite...
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semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the results obtained in the application of semidefinite programming to satisfiability and maximum-satisfiability problems. The approaches presented in some detail include the ground-breaking approximation algorithm of Goemans and Williamson for MAX-2-SAT, the Gap relaxation of de Klerk, van Maaren and Warners, and strengthenings of the Gap relaxation based on the Lasserre hierarchy of semidefinite liftings for polynomial optimization problems. We include theoretical and computational comparisons of the aforementioned semidefinite relaxations for the special case of 3-SAT, and conclude with a review of the most recent results in the application of semidefinite programming to SAT and MAX-SAT.
The minimum sum-of-squares clustering (MSSC), or kappa-means type clustering, is traditionally considered an unsupervised learning task. In recent years, the use of background knowledge to improve the cluster quality ...
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The minimum sum-of-squares clustering (MSSC), or kappa-means type clustering, is traditionally considered an unsupervised learning task. In recent years, the use of background knowledge to improve the cluster quality and promote interpretability of the clustering process has become a hot research topic at the intersection of mathematical optimization and machine learning research. The problem of taking advantage of background information in data clustering is called semi-supervised or constrained clustering. In this paper, we present branch-and-cut algorithm for semi-supervised MSSC, where background knowledge is incorporated as pairwise must-link and cannot-link constraints. For the lower bound procedure, we solve the semidefinite programming relaxation of the MSSC discrete optimization model, and we use a cutting-plane procedure for strengthening the bound. For the upper bound, instead, by using integer programming tools, we use an adaptation of the kappa-means algorithm to the constrained case. For the first time, the proposed global optimization algorithm efficiently manages to solve real-world instances up to 800 data points with different combinations of must-link and cannot-link constraints and with a generic number of features. This problem size is about four times larger than the one of the instances solved by state-of-the-art exact algorithms.
We consider the collaborative use of amplify-and-forward relays to forma beamforming system and provide physical layer security for a wireless machine-to-machine (M2M) communication network. We investigate two objecti...
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We consider the collaborative use of amplify-and-forward relays to forma beamforming system and provide physical layer security for a wireless machine-to-machine (M2M) communication network. We investigate two objectives: (i) the achievable secrecy rate maximization subject to the relay power constraint and (ii) the relay transmit power minimization under a secrecy rate constraint. For the first objective, we propose a secrecy rate maximization (SRM) beamforming scheme. The secrecy rate maximization problem can be formed into a two-level optimization problem and we solve it using semidefinite relaxation (SDR) techniques. To reduce the complexity of the SRM beamforming scheme, a virtual eavesdropper-based SRM (VE-SRM) beamforming scheme is proposed, in which we hypothesize a virtual eavesdropper instead of all eavesdroppers and maximize the secrecy rate according to the virtual eavesdropper. In addition, for the second objective, we design a relay power minimization (RPM) beamforming scheme, in which an iterative algorithm combining the SDR technology and the gradient-based method is devised by studying the convexity of the RPM problem. By relaxing the constraints of the RPM beamforming scheme, we propose a virtual eavesdropper-based RPM (VERPM) beamforming scheme, which reduces the multivariate RPM problem to a problem of a single variable, and thus an analytical solution is obtained. Our proposed beamforming designs can work well even if the number of eavesdroppers is larger than that of relays, while the existing schemes, for example, the null-space beamforming schemes, cannot work under this condition. Simulation results are presented to demonstrate the significance of performance improvements with the SRM and RPM beamforming schemes. It is also shown that the virtual eavesdropper approaches significantly reduce the complexity with acceptable performance degradation.
Inverse Optimal Control (IOC) is a powerful framework for learning a behavior from observations of experts. The framework aims to identify the underlying cost function that the observed optimal trajectories (the exper...
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Inverse Optimal Control (IOC) is a powerful framework for learning a behavior from observations of experts. The framework aims to identify the underlying cost function that the observed optimal trajectories (the experts' behavior) are optimal with respect to. In this work, we considered the case of identifying the cost and the feedback law from observed trajectories generated by an "average cost per stage"linear quadratic regulator. We show that identifying the cost is in general an ill-posed problem, and give necessary and sufficient conditions for non-identifiability. Moreover, despite the fact that the problem is in general ill-posed, we construct an estimator for the cost function and show that the control gain corresponding to this estimator is a statistically consistent estimator for the true underlying control gain. In fact, the constructed estimator is based on convex optimization, and hence the proved statistical consistency is also observed in practice. We illustrate the latter by applying the method on a simulation example from rehabilitation robotics.
Optimal experiment design for system identification involves determining an optimal input that is used to perturb the system so that the resulting input-output data is maximally informative. Plant friendly identificat...
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Optimal experiment design for system identification involves determining an optimal input that is used to perturb the system so that the resulting input-output data is maximally informative. Plant friendly identification requires that constraints on input move sizes, output sizes or variance and experiment time be respected. The solution to the optimum input design problem depends on the unknown parameters to be estimated which is often approximated by an initial estimate. Use of the estimate is likely to result in loss in performance or violation of the constraints. An alternative is to formulate a robust optimization problem with uncertain parameters. The contribution of this work is to use the uncertainty sets originating from a prior identification exercise to solve a robust plant friendly input design problem. The methodology is derived for a general class of systems illustrated using numerical simulations. Simulations validate the expectation that the constraints are probabilistically more likely to be satisfied using the robust design than a nominal design based on uncertain parameters.
In this paper we consider the problem of modelling observed data using a class of multivariate models with unknown-but-bounded (ubb) noise and uncertainty. Standard ARX models with additive and multiplicative bounded ...
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In this paper we consider the problem of modelling observed data using a class of multivariate models with unknown-but-bounded (ubb) noise and uncertainty. Standard ARX models with additive and multiplicative bounded noise belong to the considered class, as well as the deterministic counterpart of ARCH models extensively used in econometrics. We outline a method to fit these models based on historical data, and discuss the issues of set-valued forecasting.
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