Topic
Convex optimization
About: Convex optimization is a research topic. Over the lifetime, 24906 publications have been published within this topic receiving 908795 citations. The topic is also known as: convex optimisation.
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TL;DR: This work takes a node-based approach to estimation of high-dimensional Gaussian graphical models corresponding to a single set of variables under several distinct conditions, and derives a set of necessary and sufficient conditions that allows the problem to decompose into independent subproblems so that the algorithm can be scaled to high- dimensional settings.
Abstract: We consider the problem of estimating high-dimensional Gaussian graphical models corresponding to a single set of variables under several distinct conditions. This problem is motivated by the task of recovering transcriptional regulatory networks on the basis of gene expression data containing heterogeneous samples, such as different disease states, multiple species, or different developmental stages. We assume that most aspects of the conditional dependence networks are shared, but that there are some structured differences between them. Rather than assuming that similarities and differences between networks are driven by individual edges, we take a node-based approach, which in many cases provides a more intuitive interpretation of the network differences. We consider estimation under two distinct assumptions: (1) differences between the K networks are due to individual nodes that are perturbed across conditions, or (2) similarities among the K networks are due to the presence of common hub nodes that are shared across all K networks. Using a row-column overlap norm penalty function, we formulate two convex optimization problems that correspond to these two assumptions. We solve these problems using an alternating direction method of multipliers algorithm, and we derive a set of necessary and sufficient conditions that allows us to decompose the problem into independent subproblems so that our algorithm can be scaled to high-dimensional settings. Our proposal is illustrated on synthetic data, a webpage data set, and a brain cancer gene expression data set.
173 citations
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TL;DR: One of the advantages of simulated annealing, in addition to avoiding poor local minima, is that in these problems it converges faster to the minima that it finds, and it is concluded that under certain general conditions, the Boltzmann-Gibbs distributions are optimal on these convex problems.
Abstract: We apply the method known as simulated annealing to the following problem in convex optimization: Minimize a linear function over an arbitrary convex set, where the convex set is specified only by a membership oracle. Using distributions from the Boltzmann-Gibbs family leads to an algorithm that needs only O*(√n) phases for instances in Rn. This gives an optimization algorithm that makes O*(n4.5) calls to the membership oracle, in the worst case, compared to the previous best guarantee of O*(n5).
The benefits of using annealing here are surprising because such problems have no local minima that are not also global minima. Hence, we conclude that one of the advantages of simulated annealing, in addition to avoiding poor local minima, is that in these problems it converges faster to the minima that it finds. We also give a proof that under certain general conditions, the Boltzmann-Gibbs distributions are optimal for annealing on these convex problems.
173 citations
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TL;DR: It is shown that the original problem is equivalent to a convex minimization problem with simple linear constraints, and a special problem of minimizing a concave quadratic function subject to finitely many convexquadratic constraints, which is also shown to be equivalents to a minimax convex problem.
Abstract: We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints. Under a suitable simultaneous diagonalization assumption (which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints, which is also shown to be equivalent to a minimax convex problem. In both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts. Special cases and applications are also discussed. We outline interior-point polynomial-time algorithms for the solution of the equivalent convex programs.
172 citations
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15 Oct 2008
TL;DR: In this article, the authors present non-differentiable and differentiable Generalized Convex Functions (Convexity and Generalized Monotonicity), Optimality and Generative Optimality (GAN), and Optimality, Generative ConveXity, and Generalization of Quadratic Functions.
Abstract: Convex Functions.- Non-Differentiable Generalized Convex Functions.- Differentiable Generalized Convex Functions.- Optimality and Generalized Convexity.- Generalized Convexity and Generalized Monotonicity.- Generalized Convexity of Quadratic Functions.- Generalized Convexity of Some Classes of Fractional Functions.- Sequential Methods for Generalized Convex Fractional Programs.- Solutions.
172 citations
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05 Dec 2016TL;DR: This work develops fast stochastic algorithms that provably converge to a stationary point for constant minibatches and proves global linear convergence rate for an interesting subclass of nonsmooth nonconvex functions, which subsumes several recent works.
Abstract: We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gradient method with constant minibatch converges to a stationary point. To tackle this issue, we develop fast stochastic algorithms that provably converge to a stationary point for constant minibatches. Furthermore, using a variant of these algorithms, we obtain provably faster convergence than batch proximal gradient descent. Our results are based on the recent variance reduction techniques for convex optimization but with a novel analysis for handling nonconvex and nonsmooth functions. We also prove global linear convergence rate for an interesting subclass of nonsmooth nonconvex functions, which subsumes several recent works.
172 citations