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Efficient sampling from a high-dimensional Gaussian distribution is an old but high-stakes issue. Vanilla Cholesky samplers imply a computational cost and memory requirements that can rapidly become prohibitive in high dimensions. To tackle these issues, multiple methods have been proposed from different communities ranging from iterative numerical linear algebra to Markov chain Monte Carlo (MCMC) approaches. Surprisingly, no complete review and comparison of these methods has been conducted. This paper aims to review all these approaches by pointing out their differences, close relations, benefits, and limitations. In addition to reviewing the state of the art, this paper proposes a unifying Gaussian simulation framework by deriving a stochastic counterpart of the celebrated proximal point algorithm in optimization. This framework offers a novel and unifying revisiting of most of the existing MCMC approaches while also extending them. Guidelines to choosing the appropriate Gaussian simulation method for a given sampling problem in high dimensions are proposed and illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

Conjugate gradient methods are important first-order optimization algorithms both in Euclidean spaces and on Riemannian manifolds. However, while various types of conjugate gradient methods have been studied in Euclidean spaces, there are relatively fewer studies for those on Riemannian manifolds (i.e., Riemannian conjugate gradient methods). This paper proposes a novel general framework that unifies existing Riemannian conjugate gradient methods such as the ones that utilize a vector transport or inverse retraction. The proposed framework also develops other methods that have not been covered in previous studies. Furthermore, conditions for the convergence of a class of algorithms in the proposed framework are clarified. Moreover, the global convergence properties of several specific types of algorithms are extensively analyzed. The analysis provides the theoretical results for some algorithms in a more general setting than the existing studies and new developments for other algorithms. Numerical experiments are performed to confirm the validity of the theoretical results. The experimental results are used to compare the performances of several specific algorithms in the proposed framework. [ABSTRACT FROM AUTHOR]

In this paper, we study smooth stochastic multilevel composition optimization problems, where the objective function is a nested composition of T functions. We assume access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle. For solving this class of problems, we propose two algorithms using moving-average stochastic estimates, and analyze their convergence to an e-stationary point of the problem. We show that the first algorithm, which is a generalization of [S. Ghadimi, A. Ruszczynski, and M. Wang, SIAM J. Optim., 30 (2020), pp. 960-979] to the T level case, can achieve a sample complexity of OT(1/ε6) by using minibatches of samples in each iteration, where OT hides constants that depend on T. By modifying this algorithm using linearized stochastic estimates of the function values, we improve the sample complexity to OT(1/ε4). This modification not only removes the requirement of having a minibatch of samples in each iteration, but also makes the algorithm parameter-free and easy to implement. To the best of our knowledge, this is the first time that such an online algorithm designed for the (un)constrained multilevel setting obtains the same sample complexity of the smooth single-level setting, under standard assumptions (unbiasedness and boundedness of the second moments) on the stochastic first-order oracle. [ABSTRACT FROM AUTHOR]

Recent papers indicate that some algorithms for constrained optimization may exhibit worst-case complexity bounds that are very similar to those of unconstrained optimization algorithms. A natural question is whether well-established practical algorithms, perhaps with small variations, may enjoy analogous complexity results. In the present paper we show that the answer is positive with respect to inexact restoration algorithms in which first-order approximations are employed for defining the subproblems. [ABSTRACT FROM AUTHOR]

This paper studies bilevel polynomial optimization. We propose a method to solve it globally by using polynomial optimization relaxations. Each relaxation is obtained from the Karush-Kuhn-Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. For KKT conditions, Lagrange multipliers are represented as polynomial or rational functions. The Moment-sum-of-squares relaxations are used to solve the polynomial optimization relaxations. Under some general assumptions, we prove the convergence of the algorithm for solving bilevel polynomial optimization problems. Numerical experiments are presented to show the efficiency of the method. [ABSTRACT FROM AUTHOR]

With the development of robotics, there are growing needs for real time motion planning. However, due to obstacles in the environment, the planning problem is highly nonconvex, which makes it difficult to achieve real time computation using existing nonconvex optimization algorithms. This paper introduces the convex feasible set algorithm, which is a fast algorithm for nonconvex optimization problems that have convex costs and nonconvex constraints. The idea is to find a convex feasible set for the original problem and iteratively solve a sequence of subproblems using the convex constraints. The feasibility and the convergence of the proposed algorithm are proved in the paper. The application of this method on motion planning for mobile robots is discussed. The simulations demonstrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Tropical geometry has been recently used to obtain new complexity results in convex optimization and game theory. In this paper, we present an application of this approach to a famous class of algorithms for linear programming, i.e., log-barrier interior point methods. We show that these methods are not strongly polynomial by constructing a family of linear programs with 3r + 1 inequalities in dimension 2r for which the number of iterations performed is in Ω(2r). The total curvature of the central path of these linear programs is also exponential in r, disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky, and Zinchenko. These results are obtained by analyzing the tropical central path, which is the piecewise linear limit of the central paths of parameterized families of classical linear programs viewed through "logarithmic glasses." This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature in a general setting. [ABSTRACT FROM AUTHOR]

This paper generalizes the optimized gradient method (OGM) [Y. Drori and M. Teboulle, Math. Program., 145 (2014), pp. 451-482], [D. Kim and J. A. Fessler, Math. Program., 159 (2016), pp. 81-107], [D. Kim and J. A. Fessler, J. Optim. Theory Appl., 172 (2017), pp. 187-205] that achieves the optimal worst-case cost function bound of first-order methods for smooth convex minimization [Y. Drori, J. Complexity, 39 (2017), pp. 1-16]. Specifically, this paper studies a generalized formulation of OGM and analyzes its worst-case rates in terms of both the function value and the norm of the function gradient. This paper also develops a new algorithm called OGM-OG that is in the generalized family of OGM and that has the best known analytical worst-case bound with rate O(1/N1.5) on the decrease of the gradient norm among fixed-step first-order methods. This paper also proves that Nesterov's fast gradient method [Y. Nesterov, Dokl. Akad. Nauk. USSR, 269 (1983), pp. 543-547], [Y. Nesterov, Math. Program., 103 (2005), pp. 127-152] has an O(1/N1.5) worst-case gradient norm rate but with constant larger than OGM-OG. The proof is based on the worst-case analysis called the Performance Estimation Problem in [Y. Drori and M. Teboulle, Math. Program., 145 (2014), pp. 451-482]. [ABSTRACT FROM AUTHOR]

There has been much recent interest in finding unconstrained local minima of smooth functions, due in part to the prevalence of such problems in machine learning and robust statistics. A particular focus is algorithms with good complexity guarantees. Second-order Newton-type methods that make use of regularization and trust regions have been analyzed from such a perspective. More recent proposals, based chiey on first-order methodology, have also been shown to enjoy optimal iteration complexity rates, while providing additional guarantees on computational cost. In this paper, we present an algorithm with favorable complexity properties that differs in two significant ways from other recently proposed methods. First, it is based on line searches only: Each step involves computation of a search direction, followed by a backtracking line search along that direction. Second, its analysis is rather straightforward, relying for the most part on the standard technique for demonstrating sufficient decrease in the objective from backtracking. In the latter part of the paper, we consider inexact computation of the search directions, using iterative methods in linear algebra: the conjugate gradient and Lanczos methods. We derive modified convergence and complexity results for these more practical methods. [ABSTRACT FROM AUTHOR]

In this paper we consider a distributed stochastic optimization problem without gradient/subgradient information for local objective functions and subject to local convex constraints. Objective functions may be nonsmooth and observed with stochastic noises, and the network for the distributed design is time-varying. By adding stochastic dithers to local objective functions and constructing randomized differences motivated by the Kiefer--Wolfowitz algorithm, we propose a distributed subgradient-free algorithm for finding the global minimizer with local observations. Moreover, we prove that the consensus of estimates and global minimization can be achieved with probability one over the time-varying network, and we obtain the convergence rate of the mean average of estimates as well. Finally, we give numerical examples to illustrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Stochastic dominance, a pairwise comparison between random variables, is an effective tool for expressing risk aversion in stochastic optimization. In this paper, we develop a family of primal-dual algorithms for optimization problems with stochastic dominance-constraints. First, we develop an offline primal-dual algorithm and bound its optimality gap as a function of the number of iterations. Then, we extend this algorithm to the online setting where only one random sample is given in each decision epoch. We give probabilistic bounds o n the optimality gap in this setting. This technique also yields an online algorithm for the stochastic dominance-constrained multiarmed bandit with partial feedback. The paper concludes by discussing a dual approach for a batch learning problem with robust stochastic dominance constraints. [ABSTRACT FROM AUTHOR]

This paper studies distributed algorithms for the nonsmooth extended monotropic optimization problem, which is a general convex optimization problem with a certain separable structure. The considered nonsmooth objective function is the sum of local objective functions as-signed to agents in a multiagent network, with local set constraints and affine equality constraints. Each agent only knows its local objective function, local set constraint, and the information ex-changed between neighbors. To solve the constrained convex optimization problem, we propose two novel distributed continuous-time subgradient-based algorithms, with projected output feedback and derivative feedback, respectively. Moreover, we prove the convergence of proposed algorithms to the optimal solutions under some mild conditions and analyze convergence rates, with the help of the techniques of variational inequalities, decomposition methods, and differential inclusions. Finally, we give an example to illustrate the efficacy of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The computation of the singular value decomposition, or SVD, has a long history with many improvements over the years, both in its implementations and algorithmically. Here, we survey the evolution of SVD algorithms for dense matrices, discussing the motivation and performance impacts of changes. There are two main branches of dense SVD methods: bidiagonalization and Jacobi. Bidiagonalization methods started with the implementation by Golub and Reinsch in Algol60, which was subsequently ported to Fortran in the EISPACK library, and was later more efficiently implemented in the LINPACK library, targeting contemporary vector machines. To address cache-based memory hierarchies, the SVD algorithm was reformulated to use Level 3 BLAS in the LAPACK library. To address new architectures, ScaLAPACK was introduced to take advantage of distributed computing, and MAGMA was developed for accelerators such as GPUs. Algorithmically, the divide and conquer and MRRR algorithms were developed to reduce the number of operations. Still, these methods remained memory bound, so two-stage algorithms were developed to reduce memory operations and increase the computational intensity, with efficient implementations in PLASMA, DPLASMA, and MAGMA. Jacobi methods started with the two-sided method of Kogbetliantz and the one-sided method of Hestenes. They have likewise had many developments, including parallel and block versions and preconditioning to improve convergence. In this paper, we investigate the impact of these changes by testing various historical and current implementations on a common, modern multicore machine and a distributed computing platform. We show that algorithmic and implementation improvements have increased the speed of the SVD by several orders of magnitude, while using up to 40 times less energy. [ABSTRACT FROM AUTHOR]

In this paper we consider the optimization of a functional F defined as the convolution of a function f with a Gaussian kernel. We propose this type of objective function for the optimization of the output of complex computational simulations, which often present some form of deterministic noise and need to be smoothed for the results to be meaningful. We introduce a derivative-free algorithm that computes trial points from the minimization of a regression model of the noisy function f over a trust region. The regression model is constructed from function values at sample points that are chosen randomly around iterates and trial points of the algorithm. The weights given to the individual sample points in the regression problem are obtained according to an adaptive multiple importance sampling strategy. This has two advantages. First, it makes it possible to reuse all noisy function values collected over the course of the optimization. Second, the resulting regression model converges to the second-order Taylor approximation of the convolution functional F. We prove that, with probability one, each limit point of the iterates is a stationary point of F. Computational experiments on a set of benchmark problems with noisy functions compare the proposed algorithm with the deterministic derivative-free trust-region method the proposed method is based on. It is demonstrated that the proposed algorithm performs similarly efficiently in the early stages of the optimization and is able to overcome convergence problems of the original method, where the algorithm might get trapped in spurious local minima induced by the noise. [ABSTRACT FROM AUTHOR]

Algorithms for computing local minima of smooth objective functions enjoy a mature theory as well as robust and effcient implementations. By comparison, the theory and practice of saddle search is destitute. In this paper, we present results for idealized versions of the dimer and gentlest ascent (GAD) saddle search algorithms that showcase the limitations of what is theoretically achievable within the current class of saddle search algorithms: (1) we present an improved estimate on the region of attraction of saddles, (2) we give explicit examples of potential energy wells from which GAD-type dynamics are unable to escape, and (3) we present a local analysis of \singular points" around which the dynamics gets trapped and prove the existence of quasi-periodic solutions. These results indicate that it is impossible to obtain globally convergent variants of dimer- and GAD-type algorithms. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem governed by a regularized fracture model using a phase-field technique. To avoid the nondifferentiability due to the irreversibility constraint on the fracture growth, the phase-field fracture model is relaxed using a penalization approach. The existence of a solution to the penalized fracture model is shown, and the existence of at least one solution for the regularized optimal control problem is established. Moreover, the linearized fracture model is considered and used to establish first order necessary conditions as well as to discuss QP-approximations to the nonlinear optimization problem. A numerical example suggests that these can be used to obtain a fast convergent algorithm. [ABSTRACT FROM AUTHOR]

This paper presents an exact algorithm for the bilevel integer linear programming (BILP) problem. The proposed algorithm, which we call the watermelon algorithm, uses a multiway disjunction cut to remove bilevel infeasible solutions from the search space, which was motivated by how watermelon seeds can be carved out by a scoop. Serving as the scoop, a polyhedron is designed to enclose as many bilevel infeasible solutions as possible, and then the complement of this polyhedron is applied to the search space as a multiway disjunction cut in a branch-and-bound framework. We have proved that the watermelon algorithm is able to solve all BILP instances finitely and correctly, providing either a global optimal solution or a certificate of infeasibility or unboundedness. Computational experiment results on two sets of small- to medium-sized instances suggest that the watermelon algorithm could be significantly more efficient than previous branch-and-bound based BILP algorithms. [ABSTRACT FROM AUTHOR]

Solving sparse linear systems is a problem that arises in many scientific applications, and sparse direct solvers are a time-consuming and key kernel for those applications and for more advanced solvers such as hybrid direct-iterative solvers. For this reason, optimizing their performance on modern architectures is critical. The preprocessing steps of sparse direct solvers|ordering and block-symbolic factorization|are two major steps that lead to a reduced amount of computation and memory and to a better task granularity to reach a good level of performance when using BLAS kernels. With the advent of GPUs, the granularity of the block computation has become more important than ever. In this paper, we present a reordering strategy that increases this block granularity. This strategy relies on block-symbolic factorization to refine the ordering produced by tools such as Metis or Scotch, but it does not impact the number of operations required to solve the problem. We integrate this algorithm in the PaStiX solver and show an important reduction of the number of off-diagonal blocks on a large spectrum of matrices. This improvement leads to an increase in efficiency of up to 20% on GPUs. [ABSTRACT FROM AUTHOR]

Most path-following algorithms for tracing a solution path of a parametric nonlinear optimization problem are only certifiably convergent under strong regularity assumptions about the problem functions. In particular, linear independence of the constraint gradients at the solutions is typically assumed, which implies unique multipliers. In this paper we propose a procedure designed to solve problems satisfying a weaker set of conditions, allowing for nonunique (but bounded) multipliers. Each iteration along the path consists of three parts: (1) a Newton corrector step for the primal and dual variables, which is obtained by solving a linear system ot equations, (2) a predictor step for the primal and dual variables, which is found as the solution of a quadratic programming problem, and (3) a jump step for the dual variables, which is found as t he solution of a linear programming problem. We present a convergence proof and demonstrate th e successful solution tracking of the algorithm numerically on a couple of illustrative examples. [ABSTRACT FROM AUTHOR]

We propose primal-dual active-set (PDAS) methods for solving large-scale instances of an important class of convex quadratic optimization problems (QPs). The iterates of the algorithms are partitions of the index set of variables, where corresponding to each partition there exist unique primal-dual variables that can be obtained by solving a (reduced) linear system. Algorithms of this type have recently received attention when solving certain QPs and linear complementarity problems since, with rapid changes in the active set estimate, they often converge in few iterations. Indeed, as discussed in this paper, convergence in a finite number of iterations is guaranteed when a basic PDAS method is employed to solve certain QPs for which a reduced Hessian of the objective function is (a perturbation of) an M-matrix. We propose three PDAS algorithms. The novelty of the algorithms is that they allow inexactness in the (reduced) linear system solves at all partitions except optimal ones. Such a feature is particularly important in large-scale settings when one employs iterative Krylov subspace methods to solve these systems. Our first algorithm is convergent when solving problems for which properties of the Hessian can be exploited to derive explicit bounds to be enforced on the (reduced) linear system residuals, whereas our second and third algorithms employ dynamic parameters to obviate the need of such explicit bounds. We prove that when applied to solve an important class of convex QPs, our algorithms converge from any initial partition. We also illustrate their practical behavior by providing the results of numerical experiments on a set of discretized optimal control problems, some of which are explicitly formulated to exhibit degeneracy. [ABSTRACT FROM AUTHOR]

An algorithm is developed for projecting a point onto a polyhedron. The algorithm solves a dual version of the projection problem and then uses the relationship between the primal and dual to recover the projection. The techniques in the paper exploit sparsity. Sparse reconstruction by separable approximation (SpaRSA) is used to approximately identify active constraints in the polyhedron, and the dual active set algorithm (DASA) is used to compute a high precision solution. A linear convergence result is established for SpaRSA that does not require the strong concavity of the dual to the projection problem, and an earlier R-linear convergence rate is strengthened to a Q-linear convergence property. An algorithmic framework is developed for combining SpaRSA with an asymptotically preferred algorithm such as DASA. It is shown that only the preferred algorithm is executed asymptotically. Numerical results are given using the polyhedra associated with the Netlib LP test set. A comparison is made to the interior point method contained in the general purpose open source software package IPOPT for nonlinear optimization, and to the commercial package CPLEX, which contains an implementation of the barrier method that is targeted to problems with the structure of the polyhedral projection problem. [ABSTRACT FROM AUTHOR]

We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range of applications in data science, where the objective is used for inducing sparsity in the solutions while the constraint set models the noise tolerance and incorporates other prior information for data fitting. To solve this class of constrained optimization problems, a common approach is the penalty method. However, there is little theory on exact penalization for problems with nonconvex and non-Lipschitz objective functions. In this paper, we study the existence of exact penalty parameters regarding local minimizers, stationary points, and epsilonε-minimizers under suitable assumptions. Moreover, we discuss a penalty method whose subproblems are solved via a nonmonotone proximal gradient method with a suitable update scheme for the penalty parameters and prove the convergence of the algorithm to a KKT point of the constrained problem. Preliminary numerical results demonstrate the efficiency of the penalty method for finding sparse solutions of underdetermined linear systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a filter active-set algorithm for the minimization problem over a product of multiple ball/sphere constraints. By making effective use of the special structure of the ball/sphere constraints, a new limited memory BFGS (L-BFGS) scheme is presented. The new L-BFGS implementation takes advantage of the sparse structure of the Jacobian of the constraints and generates curvature information of the minimization problem. At each iteration, only two or three reduced linear systems are required to solve for the search direction. The filter technique combined with the backtracking line search strategy ensures the global convergence, and the local superlinear convergence can also be established under mild conditions. The algorithm is applied to two specific applications, the nearest correlation matrix with factor structure and the maximal correlation problem. Our numerical experiments indicate that the proposed algorithm is competitive with some recently custom-designed methods for each individual application. [ABSTRACT FROM AUTHOR]

In this paper, we present a new conditional gradient type method for convex optimization by calling a linear optimization (LO) oracle to minimize a series of linear functions over the feasible set. Different from the classic conditional gradient method, the conditional gradient sliding (CGS) algorithm developed herein can skip the computation of gradients from time to time and, as a result, can achieve the optimal complexity bounds in terms of not only the number of calls to the LO oracle but also the number of gradient evaluations. More specifically, we show that the CGS method requires O(1/√ε} and O(log (1/ε) gradient evaluations, respectively, for solving smooth and strongly convex problems, while still maintaining the optimal O(1/ε) bound on the number of calls to the LO oracle. We also develop variants of the CGS method which can achieve the optimal complexity bounds for solving stochastic optimization problems and an important class of saddle point optimization problems. To the best of our knowledge, this is the first time that these types of projection-free optimal first-order methods have been developed in the literature. Some preliminary numerical results have also been provided to demonstrate the advantages of the CGS method. [ABSTRACT FROM AUTHOR]

This paper is about distributed derivative-based algorithms for solving optimization problems with a separable (potentially nonconvex) objective function and coupled ane constraints. A parallelizable method is proposed that combines ideas from the fields of sequential quadratic programming and augmented Lagrangian algorithms. The method negotiates shared dual variables that may be interpreted as prices, a concept employed in dual decomposition methods and the alternating direction method of multipliers (ADMM). Here, each agent solves its own small-scale nonlinear programming problem and communicates with other agents by solving coupled quadratic programming problems. These coupled quadratic programming problems have equality constraints for which parallelizable methods are available. The use of techniques associated with standard sequential quadratic programming methods gives a method with superlinear or quadratic convergence rate under suitable conditions. This is in contrast to existing decomposition methods, such as ADMM, which have a linear convergence rate. It is shown how the proposed algorithm may be extended using globalization techniques that guarantee convergence to a local minimizer from any initial starting point. [ABSTRACT FROM AUTHOR]

This paper studies the robustness properties against additive persistent noise of a class of distributed continuous-time algorithms for convex optimization. A team of agents, each with its own private objective function, seeks to collectively determine the global decision vector that minimizes the sum of all the objectives. The team communicates over a weight-balanced, strongly connected digraph and both interagent communication and agent computation are corrupted by noise. Under the proposed distributed algorithm, each agent updates its estimate of the global solution using the gradient of its local objective function while, at the same time, seeking to agree with its neighbors' estimates via proportional-integral feedback on their disagreement. Under mild conditions on the local objective functions, we show that this strategy is noise-to-state exponentially stable in the second moment with respect to the optimal solution. Our technical approach combines notions and tools from graph theory, stochastic differential equations, Lyapunov stability analysis, and cocoercivity of vector fields. Simulations illustrate our results. [ABSTRACT FROM AUTHOR]

This paper develops a new framework to analyze and design iterative optimization algorithms built on the notion of integral quadratic constraints (IQCs) from robust control theory. IQCs provide sufficient conditions for the stability of complicated interconnected systems, and these conditions can be checked by semidefinite programming. We discuss how to adapt IQC theory to study optimization algorithms, proving new inequalities about convex functions and providing a version of IQC theory adapted for use by optimization researchers. Using these inequalities, we derive numerical upper bounds on convergence rates for the Gradient method, the Heavy-ball method, Nesterov's accelerated method, and related variants by solving small, simple semidefinite programming problems. We also briefly show how these techniques can be used to search for optimization algorithms with desired performance characteristics, establishing a new methodology for algorithm design. [ABSTRACT FROM AUTHOR]

This paper proposes a new algorithm for solving a class of composite convex-concave saddle-point problems. The new algorithm is a special instance of the hybrid proximal extragradient framework in which a Nesterov accelerated variant is used to approximately solve the prox subproblems. One of the advantages of the new method is that it works for any constant choice of proximal stepsize. Moreover, a suitable choice of the latter stepsize yields a method with the best known (accelerated inner) iteration complexity for the aforementioned class of saddle-point problems. In contrast to the smoothing technique of [Y. Nesterov, Math. Program., 103 (2005), pp. 127-152], our accelerated method does not assume that a feasible set is bounded due to its proximal point nature. Experiment results on three problem sets show that the new method outperforms Nesterov's smoothing technique of [Y. Nesterov, Math. Program., 103 (2005), pp. 127-152]. [ABSTRACT FROM AUTHOR]

In this paper we study the convex problem of optimizing the sum of a smooth function and a compactly supported nonsmooth term with a specific separable form. We analyze the block version of the generalized conditional gradient method when the blocks are chosen in a cyclic order. A global sublinear rate of convergence is established for two different stepsize strategies commonly used in this class of methods. Numerical comparisons of the proposed method to both the classical conditional gradient algorithm and its random block version demonstrate the effectiveness of the cyclic block update rule. [ABSTRACT FROM AUTHOR]

Trevisan [SIAM J. Comput., 41 (2012), pp. 1769-1786] presented an algorithm for Max-Cut based on spectral partitioning techniques. This is the first algorithm for Max-Cut with an approximation guarantee strictly larger than 1/2 that is not based on semidefinite programming. Trevisan showed that its approximation ratio is of at least 0.531. In this paper we improve this bound up to 0.614247. We also define and extend this result for the more general maximum colored cut problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety M≤κ of real m×n matrices of rank at most k. Such methods extend Riemannian optimization methods, which are successfully used on the smooth manifold Mk of rankk matrices, to its closure by taking steps along gradient-related directions in the tangent cone, and afterwards projecting back to M≤κ. Considering such a method circumvents the difficulties which arise from the nonclosedness and the unbounded curvature of Mk. The pointwise convergence is obtained for real-analytic functions on the basis of a Lojasiewicz inequality for the projection of the antigradient to the tangent cone. If the derived limit point lies on the smooth part of M≤κ, i.e., in Mk, this boils down to more or less known results, but with the benefit that asymptotic convergence rate estimates (for specific step-sizes) can be obtained without an a priori curvature bound, simply from the fact that the limit lies on a smooth manifold. At the same time, one can give a convincing justification for assuming critical points to lie in Mk: if X is a critical point of f on M≤κ, then either X has rank k, or ∇f(X) = 0. [ABSTRACT FROM AUTHOR]

The problem of data representation on a sphere of unknown radius arises from various disciplines such as statistics (spatial data representation), psychology (constrained multidimensional scaling), and computer science (machine learning and pattern recognition). The best representation often needs to minimize a distance function of the data on a sphere as well as to satisfy some Euclidean distance constraints. It is those spherical and Euclidean distance constraints that present an enormous challenge to the existing algorithms. In this paper, we reformulate the problem as an Euclidean distance matrix optimization problem with a low rank constraint. We then propose an iterative algorithm that uses a quadratically convergent Newton-CG method at each step. We study fundamental issues including constraint nondegeneracy and the nonsingularity of generalized Jacobian that ensure the quadratic convergence of the Newton method. We use some classic examples from the spherical multidimensional scaling to demonstrate the flexibility of the algorithm in incorporating various constraints. We also present an interesting application to the circle fitting problem. [ABSTRACT FROM AUTHOR]

Given a convex optimization problem and its dual, there are many possible firstorder algorithms. In this paper, we show the equivalence between mirror descent algorithms and algorithms generalizing the conditional gradient method. This is done through convex duality and implies notably that for certain problems, such as for supervised machine learning problems with nonsmooth losses or problems regularized by nonsmooth regularizers, the primal subgradient method and the dual conditional gradient method are formally equivalent. The dual interpretation leads to a form of line search for mirror descent, as well as guarantees of convergence for primal-dual certificates. [ABSTRACT FROM AUTHOR]

We present an improved version of an infeasible interior-point method for linear optimization published in 2006. In the earlier version each iteration consisted of one so-called feasibility step and a few--at most three--centering steps. In this paper each iteration consists of only a feasibility step, whereas the iteration bound improves the earlier bound by a factor 2 √2. The improvements are due to a new lemma that gives a much tighter upper bound for the proximity after the feasibility step. [ABSTRACT FROM AUTHOR]