124,782 research outputs found

    Self-scaled barriers for irreducible symmetric cones

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    Self-scaled barrier functions are fundamental objects in the theory of interior-point methods for linear optimization over symmetric cones, of which linear and semidefinite programming are special cases. We are classifying all self-scaled barriers over irreducible symmetric cones and show that these functions are merely homothetic transformations of the universal barrier function. Together with a decomposition theorem for self-scaled barriers this concludes the algebraic classification theory of these functions. After introducing the reader to the concepts relevant to the problem and tracing the history of the subject, we start by deriving our result from first principles in the important special case of semidefinite programming. We then generalise these arguments to irreducible symmetric cones by invoking results from the theory of Euclidean Jordan algebras.Comment: 12 page

    Generalized Self-concordant Hessian-barrier algorithms

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    Many problems in statistical learning, imaging, and computer vision involve the optimization of a non-convex objective function with singularities at the boundary of the feasible set. For such challenging instances, we develop a new interior-point technique building on the Hessian-barrier algorithm recently introduced in Bomze, Mertikopoulos, Schachinger and Staudigl, [SIAM J. Opt. 2019 29(3), pp. 2100-2127], where the Riemannian metric is induced by a generalized self-concordant function. This class of functions is sufficiently general to include most of the commonly used barrier functions in the literature of interior point methods. We prove global convergence to an approximate stationary point of the method, and in cases where the feasible set admits an easily computable self-concordant barrier, we verify worst-case optimal iteration complexity of the method. Applications in non-convex statistical estimation and LpL^{p}-minimization are discussed to given the efficiency of the method

    Generalized self-concordant Hessian-barrier algorithms

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    Many problems in statistical learning, imaging, and computer vision involve the optimization of a non-convex objective function with singularities at the boundary of the feasible set. For such challenging instances, we develop a new interior-point technique building on the Hessian-barrier algorithm recently introduced in Bomze, Mertikopoulos, Schachinger and Staudigl, [SIAM J. Opt. 2019 29(3), pp. 2100-2127], where the Riemannian metric is induced by a generalized selfconcordant function. This class of functions is sufficiently general to include most of the commonly used barrier functions in the literature of interior point methods. We prove global convergence to an approximate stationary point of the method, and in cases where the feasible set admits an easily computable self-concordant barrier, we verify worst-case optimal iteration complexity of the method. Applications in non-convex statistical estimation and Lp-minimization are discussed to given the efficiency of the method

    Primal-Dual Algorithms for Semidefinit Optimization Problems based on generalized trigonometric barrier function

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    Recently, M. Bouafoa, et al. (Journal of optimization Theory and Applications, August, 2016), investigated a new kernel function which differs from the self-regular kernel functions. The kernel function has a trigonometric Barrier Term. In this paper we generalize the analysis presented in the above paper for Semidefinit Optimization Problems (SDO). It is shown that the interior-point methods based on this function for large-update methods, the iteration bound is improved significantly. For small-update interior point methods the iteration bound is the best currently known bound for primal-dual interior point methods. The analysis for SDO deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper.publishedVersio

    Kernel-Based Interior-Point Methods for Cartesian \u3cem\u3eP\u3c/em\u3e*(Îș)-Linear Complementarity Problems over Symmetric Cones

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    We present an interior point method for Cartesian P*(k)-Linear Complementarity Problems over Symmetric Cones (SCLCPs). The Cartesian P*(k)-SCLCPs have been recently introduced as the generalization of the more commonly known and more widely used monotone SCLCPs. The IPM is based on the barrier functions that are defined by a large class of univariate functions called eligible kernel function which have recently been successfully used to design new IPMs for various optimization problems. Eligible barrier (kernel) functions are used in calculating the Nesterov-Todd search directions and the default step-size which leads to a very good complexity results for the method. For some specific eligilbe kernel functions we match the best known iteration bound for the long-step methods while for the short-step methods the best iteration bound is matched for all cases

    Local quadratic convergence of polynomial-time interior-point methods for conic optimization problems

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    In this paper, we establish a local quadratic convergence of polynomial-time interior-point methods for general conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier functions. We propose new path-following predictor-corrector schemes which work only in the dual space. They are based on an easily computable gradient proximity measure, which ensures an automatic transformation of the global linear rate of convergence to the local quadratic one under some mild assumptions. Our step-size procedure for the predictor step is related to the maximum step size (the one that takes us to the boundary). It appears that in order to obtain local superlinear convergence, we need to tighten the neighborhood of the central path proportionally to the current duality gapconic optimization problem, worst-case complexity analysis, self-concordant barriers, polynomial-time methods, predictor-corrector methods, local quadratic convergence

    Interior-point methods for P∗(Îș)-linear complementarity problem based on generalized trigonometric barrier function

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    Recently, M.~Bouafoa, et al. investigated a new kernel function which differs from the self-regular kernel functions. The kernel function has a trigonometric Barrier Term. In this paper we generalize the analysis presented in the above paper for P∗(Îș)P_{*}(\kappa) Linear Complementarity Problems (LCPs). It is shown that the iteration bound for primal-dual large-update and small-update interior-point methods based on this function is as good as the currently best known iteration bounds for these type methods. The analysis for LCPs deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper.publishedVersio

    Interior Point Methods with a Gradient Oracle

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    We provide an interior point method based on quasi-Newton iterations, which only requires first-order access to a strongly self-concordant barrier function. To achieve this, we extend the techniques of Dunagan-Harvey [STOC '07] to maintain a preconditioner, while using only first-order information. We measure the quality of this preconditioner in terms of its relative excentricity to the unknown Hessian matrix, and we generalize these techniques to convex functions with a slowly-changing Hessian. We combine this with an interior point method to show that, given first-order access to an appropriate barrier function for a convex set KK, we can solve well-conditioned linear optimization problems over KK to Δ\varepsilon precision in time O~((T+n2)nÎœlog⁥(1/Δ))\widetilde{O}\left(\left(\mathcal{T}+n^{2}\right)\sqrt{n\nu}\log\left(1/\varepsilon\right)\right), where Îœ\nu is the self-concordance parameter of the barrier function, and T\mathcal{T} is the time required to make a gradient query. As a consequence we show that: ∙\bullet Linear optimization over nn-dimensional convex sets can be solved in time O~((Tn+n3)log⁥(1/Δ))\widetilde{O}\left(\left(\mathcal{T}n+n^{3}\right)\log\left(1/\varepsilon\right)\right). This parallels the running time achieved by state of the art algorithms for cutting plane methods, when replacing separation oracles with first-order oracles for an appropriate barrier function. ∙\bullet We can solve semidefinite programs involving m≄nm\geq n matrices in Rn×n\mathbb{R}^{n\times n} in time O~(mn4+m1.25n3.5log⁥(1/Δ))\widetilde{O}\left(mn^{4}+m^{1.25}n^{3.5}\log\left(1/\varepsilon\right)\right), improving over the state of the art algorithms, in the case where m=Ω(n3.5ω−1.25)m=\Omega\left(n^{\frac{3.5}{\omega-1.25}}\right). Along the way we develop a host of tools allowing us to control the evolution of our potential functions, using techniques from matrix analysis and Schur convexity.Comment: STOC 202

    Hessian barrier algorithms for linearly constrained optimization problems

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    In this paper, we propose an interior-point method for linearly constrained optimization problems (possibly nonconvex). The method - which we call the Hessian barrier algorithm (HBA) - combines a forward Euler discretization of Hessian Riemannian gradient flows with an Armijo backtracking step-size policy. In this way, HBA can be seen as an alternative to mirror descent (MD), and contains as special cases the affine scaling algorithm, regularized Newton processes, and several other iterative solution methods. Our main result is that, modulo a non-degeneracy condition, the algorithm converges to the problem's set of critical points; hence, in the convex case, the algorithm converges globally to the problem's minimum set. In the case of linearly constrained quadratic programs (not necessarily convex), we also show that the method's convergence rate is O(1/kρ)\mathcal{O}(1/k^\rho) for some ρ∈(0,1]\rho\in(0,1] that depends only on the choice of kernel function (i.e., not on the problem's primitives). These theoretical results are validated by numerical experiments in standard non-convex test functions and large-scale traffic assignment problems.Comment: 27 pages, 6 figure
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