Search

In this work, we consider convex optimization problems with smooth objective function and nonsmooth functional constraints. We propose a new stochastic gradient algorithm, called Stochastic Halfspace Approximation Method (SHAM), to solve this problem, where at each iteration we first take a gradient step for the objective function and then we perform a projection step onto one halfspace approximation of a randomly chosen constraint. We propose various strategies to create this stochastic halfspace approximation and we provide a unified convergence analysis that yields new convergence rates for SHAM algorithm in both optimality and feasibility criteria evaluated at some average point. In particular, we derive convergence rates of order $\mathcal{O} (1/\sqrt{k})$, when the objective function is only convex, and $\mathcal{O} (1/k)$ when the objective function is strongly convex. The efficiency of SHAM is illustrated through detailed numerical simulations.

Dimensionality reduction can be applied to hyperspectral images so that the most useful data can be extracted and processed more quickly. This is critical in any situation in which data volume exceeds the capacity of the computational resources, particularly in the case of remote sensing platforms (e.g., drones, satellites), but also in the case of multi-year datasets. Moreover, the computational strategies of unsupervised dimensionality reduction often provide the basis for more complicated supervised techniques. Seven unsupervised dimensionality reduction algorithms are tested on hyperspectral data from the HYPSO-1 earth observation satellite. Each particular algorithm is chosen to be representative of a broader collection. The experiments probe the computational complexity, reconstruction accuracy, signal clarity, sensitivity to artifacts, and effects on target detection and classification of the different algorithms. No algorithm consistently outperformed the others across all tests, but some general trends regarding the characteristics of the algorithms did emerge. With half a million pixels, computational time requirements of the methods varied by 5 orders of magnitude, and the reconstruction error varied by about 3 orders of magnitude. A relationship between mutual information and artifact susceptibility was suggested by the tests. The relative performance of the algorithms differed significantly between the target detection and classification tests. Overall, these experiments both show the power of dimensionality reduction and give guidance regarding how to evaluate a technique prior to incorporating it into a processing pipeline., Comment: 16 pages, 11 figures

In this paper we develop a higher-order method for solving composite (non)convex minimization problems with smooth (non)convex functional constraints. At each iteration our method approximates the smooth part of the objective function and of the constraints by higher-order Taylor approximations, leading to a moving Taylor approximation method (MTA). We present convergence guarantees for MTA algorithm for both, nonconvex and convex problems. In particular, when the objective and the constraints are nonconvex functions, we prove that the sequence generated by MTA algorithm converges globally to a KKT point. Moreover, we derive convergence rates in the iterates when the problem data satisfy the Kurdyka-Lojasiewicz (KL) property. Further, when the objective function is (uniformly) convex and the constraints are also convex, we provide (linear/superlinear) sublinear convergence rates for our algorithm. Finally, we present an efficient implementation of the proposed algorithm and compare it with existing methods from the literature., Comment: 26 pages, June 2023

Hyperspectral Imaging comprises excessive data consequently leading to significant challenges for data processing, storage and transmission. Compressive Sensing has been used in the field of Hyperspectral Imaging as a technique to compress the large amount of data. This work addresses the recovery of hyperspectral images 2.5x compressed. A comparative study in terms of the accuracy and the performance of the convex FISTA/ADMM in addition to the greedy gOMP/BIHT/CoSaMP recovery algorithms is presented. The results indicate that the algorithms recover successfully the compressed data, yet the gOMP algorithm achieves superior accuracy and faster recovery in comparison to the other algorithms at the expense of high dependence on unknown sparsity level of the data to recover., Comment: Hyperspectral Imaging, Compressive Sensing, Convex Algorithms, Greedy Algorithms, FISTA, ADMM, gOMP, BIHT, CoSaMP, IEEE-copyrighted material (2022), IVMSP Workshop (26-29 June 2022)

We introduce the concept of inexact first-order oracle of degree q for a possibly nonconvex and nonsmooth function, which naturally appears in the context of approximate gradient, weak level of smoothness and other situations. Our definition is less conservative than those found in the existing literature, and it can be viewed as an interpolation between fully exact and the existing inexact first-order oracle definitions. We analyze the convergence behavior of a (fast) inexact proximal gradient method using such an oracle for solving (non)convex composite minimization problems. We derive complexity estimates and study the dependence between the accuracy of the oracle and the desired accuracy of the gradient or of the objective function. Our results show that better rates can be obtained both theoretically and in numerical simulations when q is large., Comment: 21 pages

In this paper we consider finite sum composite convex optimization problems with many functional constraints. The objective function is expressed as a finite sum of two terms, one of which admits easy computation of (sub)gradients while the other is amenable to proximal evaluations. We assume a generalized bounded gradient condition on the objective which allows us to simultaneously tackle both smooth and nonsmooth problems. We also consider the cases of both with and without a strong convexity property. Further, we assume that each constraint set is given as the level set of a convex but not necessarily differentiable function. We reformulate the constrained finite sum problem into a stochastic optimization problem for which the stochastic subgradient projection method from [17] specializes to a collection of mini-batch variants, with different mini-batch sizes for the objective function and functional constraints, respectively. More specifically, at each iteration, our algorithm takes a mini-batch stochastic proximal subgradient step aimed at minimizing the objective function and then a subsequent mini-batch subgradient projection step minimizing the feasibility violation. By specializing different mini-batching strategies, we derive exact expressions for the stepsizes as a function of the mini-batch size and in some cases we also derive insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime. We also prove sublinear convergence rates for the mini-batch subgradient projection algorithm which depend explicitly on the mini-batch sizes and on the properties of the objective function. Numerical results also show a better performance of our mini-batch scheme over its single-batch counterpart., Comment: 24 pages, corrections November 2024

Deep neural networks have revolutionized many fields, including image processing, inverse problems, text mining and more recently, give very promising results in systems and control. Neural networks with hidden layers have a strong potential as an approximation framework of predictive control laws as they usually yield better approximation quality and smaller memory requirements than existing explicit (multi-parametric) approaches. In this paper, we first show that neural networks with HardTanh activation functions can exactly represent predictive control laws of linear time-invariant systems. We derive theoretical bounds on the minimum number of hidden layers and neurons that a HardTanh neural network should have to exactly represent a given predictive control law. The choice of HardTanh deep neural networks is particularly suited for linear predictive control laws as they usually require less hidden layers and neurons than deep neural networks with ReLU units for representing exactly continuous piecewise affine (or equivalently min-max) maps. In the second part of the paper we bring the physics of the model and standard optimization techniques into the architecture design, in order to eliminate the disadvantages of the black-box HardTanh learning. More specifically, we design trainable unfolded HardTanh deep architectures for learning linear predictive control laws based on two standard iterative optimization algorithms, i.e., projected gradient descent and accelerated projected gradient descent. We also study the performance of the proposed HardTanh type deep neural networks on a linear model predictive control application., Comment: 10 pages

Higher-order tensor methods were recently proposed for minimizing smooth convex and nonconvex functions. Higher-order algorithms accelerate the convergence of the classical first-order methods thanks to the higher-order derivatives used in the updates. The purpose of this paper is twofold. Firstly, to show that the higher-order algorithmic framework can be generalized and successfully applied to (nonsmooth) difference of convex functions, namely, those that can be expressed as the difference of two smooth convex functions and a possibly nonsmooth convex one. We also provide examples when the subproblem can be solved efficiently, even globally. Secondly, to derive a complete convergence analysis for our higher-order difference of convex functions (HO-DC) algorithm. In particular, we prove that any limit point of the HO-DC iterative sequence is a critical point of the problem under consideration, the corresponding objective value is monotonically decreasing and the minimum value of the norms of its subgradients converges globally to zero at a sublinear rate. The sublinear or linear convergence rates of the iterations are obtained under the Kurdyka-Lojasiewicz property., Comment: 17 pages

This paper deals with convex nonsmooth optimization problems. We introduce a general smooth approximation framework for the original function and apply random (accelerated) coordinate descent methods for minimizing the corresponding smooth approximations. Our framework covers the most important classes of smoothing techniques from the literature. Based on this general framework for the smooth approximation and using coordinate descent type methods we derive convergence rates in function values for the original objective. Moreover, if the original function satisfies a growth condition, then we prove that the smooth approximations also inherits this condition and consequently the convergence rates are improved in this case. We also present a relative randomized coordinate descent algorithm for solving nonseparable minimization problems with the objective function relative smooth along coordinates w.r.t. a (possibly nonseparable) differentiable function. For this algorithm we also derive convergence rates in the convex case and under the growth condition for the objective., Comment: 40 pages

Optimizing strongly convex functions subject to linear constraints is a fundamental problem with numerous applications. In this work, we propose a block (accelerated) randomized Bregman-Kaczmarz method that only uses a block of constraints in each iteration to tackle this problem. We consider a dual formulation of this problem in order to deal in an efficient way with the linear constraints. Using convex tools, we show that the corresponding dual function satisfies the Polyak-Lojasiewicz (PL) property, provided that the primal objective function is strongly convex and verifies additionally some other mild assumptions. However, adapting the existing theory on coordinate descent methods to our dual formulation can only give us sublinear convergence results in the dual space. In order to obtain convergence results in some criterion corresponding to the primal (original) problem, we transfer our algorithm to the primal space, which combined with the PL property allows us to get linear convergence rates. More specifically, we provide a theoretical analysis of the convergence of our proposed method under different assumptions on the objective and demonstrate in the numerical experiments its superior efficiency and speed up compared to existing methods for the same problem.

In this paper, we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints are locally smooth. For solving this problem, we propose a linearized augmented Lagrangian method, i.e., we linearize the functional constraints in the augmented Lagrangian at the current iterate and add a quadratic regularization, yielding a subproblem that is easy to solve, and whose solution is the next iterate. Under a dynamic regularization parameter choice, we prove global asymptotic convergence of the iterates to a critical point of the problem. We also derive convergence guarantees for the iterates of our method to an $\epsilon$ first-order optimal solution in $\mathcal{O}(1/{\epsilon^2})$ outer iterations. Finally, we show that, when the problem data are e.g., semialgebraic, the sequence generated by our algorithm converges and we derive convergence rates. We validate the theory and the performance of the proposed algorithm by numerically comparing it with the existing methods from the literature., Comment: 32

In this paper we consider convex optimization problems with stochastic composite objective function subject to (possibly) infinite intersection of constraints. The objective function is expressed in terms of expectation operator over a sum of two terms satisfying a stochastic bounded gradient condition, with or without strong convexity type properties. In contrast to the classical approach, where the constraints are usually represented as intersection of simple sets, in this paper we consider that each constraint set is given as the level set of a convex but not necessarily differentiable function. Based on the flexibility offered by our general optimization model we consider a stochastic subgradient method with random feasibility updates. At each iteration, our algorithm takes a stochastic proximal (sub)gradient step aimed at minimizing the objective function and then a subsequent subgradient step minimizing the feasibility violation of the observed random constraint. We analyze the convergence behavior of the proposed algorithm for diminishing stepsizes and for the case when the objective function is convex or strongly convex, unifying the nonsmooth and smooth cases. We prove sublinear convergence rates for this stochastic subgradient algorithm, which are known to be optimal for subgradient methods on this class of problems. When the objective function has a linear least-square form and the constraints are polyhedral, it is shown that the algorithm converges linearly. Numerical evidence supports the effectiveness of our method in real problems., Comment: 32 pages, initial version July 2021, corrections November 2024

In this paper we consider large-scale composite optimization problems having the objective function formed as a sum of two terms (possibly nonconvex), one has (block) coordinate-wise Lipschitz continuous gradient and the other is differentiable but nonseparable. Under these general settings we derive and analyze two new coordinate descent methods. The first algorithm, referred to as coordinate proximal gradient method, considers the composite form of the objective function, while the other algorithm disregards the composite form of the objective and uses the partial gradient of the full objective, yielding a coordinate gradient descent scheme with novel adaptive stepsize rules. We prove that these new stepsize rules make the coordinate gradient scheme a descent method, provided that additional assumptions hold for the second term in the objective function. We present a complete worst-case complexity analysis for these two new methods in both, convex and nonconvex settings, provided that the (block) coordinates are chosen random or cyclic. Preliminary numerical results also confirm the efficiency of our two algorithms on practical problems., Comment: 30 pages. arXiv admin note: text overlap with arXiv:2104.13370

This paper deals with composite optimization problems having the objective function formed as the sum of two terms, one has Lipschitz continuous gradient along random subspaces and may be nonconvex and the second term is simple and differentiable, but possibly nonconvex and nonseparable. Under these settings we design a stochastic coordinate proximal gradient method which takes into account the nonseparable composite form of the objective function. This algorithm achieves scalability by constructing at each iteration a local approximation model of the whole nonseparable objective function along a random subspace with user-determined dimension. We outline efficient techniques for selecting the random subspace, yielding an implementation that has low cost per-iteration while also achieving fast convergence rates. We present a probabilistic worst-case complexity analysis for our stochastic coordinate proximal gradient method in convex and nonconvex settings, in particular we prove high-probability bounds on the number of iterations before a given optimality is achieved. Extensive numerical results also confirm the efficiency of our algorithm., Comment: 31 pages

Majorization-minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function so that along the iterations the objective function decreases. Such a simple principle allows to solve a large class of optimization problems, even nonconvex and nonsmooth. We propose a general higher-order majorization-minimization algorithmic framework for minimizing an objective function that admits an approximation (surrogate) such that the corresponding error function has a higher-order Lipschitz continuous derivative. We present convergence guarantees for our new method for general optimization problems with (non)convex and/or (non)smooth objective function. For convex (possibly nonsmooth) problems we provide global sublinear convergence rates, while for problems with uniformly convex objective function we obtain locally faster superlinear convergence rates. We also prove global stationary point guarantees for general nonconvex (possibly nonsmooth) problems and under Kurdyka-Lojasiewicz property of the objective function we derive local convergence rates ranging from sublinear to superlinear for our majorization-minimization algorithm. Moreover, for unconstrained nonconvex problems we derive convergence rates in terms of first- and second-order optimality conditions., Comment: 30 pages

In this paper, we compute a low order approximation of a system of large order $n$ that matches $\nu$ moments of order $j_i$ of the transfer function, at $\nu$ interpolation points, has $\ell$ poles and $k$ zeros fixed and also matches $\nu-(\ell +k)$ moments of order $j_i+1$, where $j_i+1$ is the multiplicity of the $i$-th interpolation point. We derive explicit linear systems in the free parameters to simultaneously achieve the required pole-zero placement and match the desired high order moments. We compute the closed form of the free parameters that meet the constraints, as the solution of a $\nu$ order linear system. Furthermore, for data-driven model reduction, we generalize the construction of the Loewner matrices to include the data and the imposed pole and higher order moment constraints. The resulting approximations achieve a trade-off between the good norm approximation and the preservation of the dynamics of the original system in a region of interest., Comment: 7 pages

In this paper we consider stochastic composite convex optimization problems with the objective function satisfying a stochastic bounded gradient condition, with or without a quadratic functional growth property. These models include the most well-known classes of objective functions analyzed in the literature: non-smooth Lipschitz functions and composition of a (potentially) non-smooth function and a smooth function, with or without strong convexity. Based on the flexibility offered by our optimization model we consider several variants of stochastic first order methods, such as the stochastic proximal gradient and the stochastic proximal point algorithms. Usually, the convergence theory for these methods has been derived for simple stochastic optimization models satisfying restrictive assumptions, the rates are in general sublinear and hold only for specific decreasing stepsizes. Hence, we analyze the convergence rates of stochastic first order methods with constant or variable stepsize under general assumptions covering a large class of objective functions. For constant stepsize we show that these methods can achieve linear convergence rate up to a constant proportional to the stepsize and under some strong stochastic bounded gradient condition even pure linear convergence. Moreover, when a variable stepsize is chosen we derive sublinear convergence rates for these stochastic first order methods. Finally, the stochastic gradient mapping and the Moreau smoothing mapping introduced in the present paper lead to simple and intuitive proofs., Comment: The results of this paper have been obtained by the author since 2017 and presented at several international conferences: e.g., Conference on Recent Advances in Artificial Intelligence, June 2017; KAUST Research Workshop on Optimization and Big Data, February 2018

This paper deals with constrained convex problems, where the objective function is smooth strongly convex and the feasible set is given as the intersection of a large number of closed convex (possibly non-polyhedral) sets. In order to deal efficiently with the complicated constraints we consider a dual formulation of this problem. We prove that the corresponding dual function satisfies a quadratic growth property on any sublevel set, provided that the objective function is smooth and strongly convex and the sets verify the Slater's condition. To the best of our knowledge, this work is the first deriving a quadratic growth condition for the dual under these general assumptions. Existing works derive similar quadratic growth conditions under more conservative assumptions, e.g., the sets need to be either polyhedral or compact. Then, for finding the minimum of the dual problem, due to its special composite structure, we propose random (accelerated) coordinate descent algorithms. However, with the existing theory one can prove that such methods converge only sublinearly. Based on our new quadratic growth property derived for the dual, we now show that such methods have faster convergence, that is the dual random (accelerated) coordinate descent algorithms converge linearly. Besides providing a general dual framework for the analysis of randomized coordinate descent schemes, our results resolve an open problem in the literature related to the convergence of Dykstra algorithm on the best feasibility problem for a collection of convex sets. That is, we establish linear convergence rate for the randomized Dykstra algorithm when the convex sets satisfy the Slater's condition and derive also a new accelerated variant for the Dykstra algorithm., Comment: 27 pages

This paper deals with the convex feasibility problem, where the feasible set is given as the intersection of a (possibly infinite) number of closed convex sets. We assume that each set is specified algebraically as a convex inequality, where the associated convex function is general (possibly non-differentiable). For finding a point satisfying all the convex inequalities we design and analyze random projection algorithms using special subgradient iterations and extrapolated stepsizes. Moreover, the iterate updates are performed based on parallel random observations of several constraint components. For these minibatch stochastic subgradient-based projection methods we prove sublinear convergence results and, under some linear regularity condition for the functional constraints, we prove linear convergence rates. We also derive conditions under which these rates depend explicitly on the minibatch size. To the best of our knowledge, this work is the first deriving conditions that show when minibatch stochastic subgradient-based projection updates have a better complexity than their single-sample variants., Comment: 12 pages

We propose a globally convergent Gauss-Newton algorithm for finding a local optimal solution of a non-convex and possibly non-smooth optimization problem. The algorithm that we present is based on a Gauss-Newton-type iteration for the non-smooth penalized formulation of the original problem. We establish a global convergence rate for this scheme from any initial point to a stationary point of the problem while using an exact penalty formulation. Under some more restrictive conditions we also derive local quadratic convergence for this scheme. We apply our proposed algorithm to solve the Alternating Current optimal power flow problem on meshed electricity networks, which is a fundamental application in power systems engineering. We verify the performance of the proposed method by showing comparable behavior with IPOPT, a well-established solver. We perform our validation on several representative instances of the optimal power flow problem, which are sourced from the MATPOWER library., Comment: 32 pages

In this paper we consider non-smooth convex optimization problems with (possibly) infinite intersection of constraints. In contrast to the classical approach, where the constraints are usually represented as intersection of simple sets, which are easy to project onto, in this paper we consider that each constraint set is given as the level set of a convex but not necessarily differentiable function. For these settings we propose subgradient iterative algorithms with random minibatch feasibility updates. At each iteration, our algorithms take a step aimed at only minimizing the objective function and then a subsequent step minimizing the feasibility violation of the observed minibatch of constraints. The feasibility updates are performed based on either parallel or sequential random observations of several constraint components. We analyze the convergence behavior of the proposed algorithms for the case when the objective function is strongly convex and with bounded subgradients, while the functional constraints are endowed with a bounded first-order black-box oracle. For a diminishing stepsize, we prove sublinear convergence rates for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected suboptimality of the function values along the weighted averages. Our convergence rates are known to be optimal for subgradient methods on this class of problems. Moreover, the rates depend explicitly on the minibatch size and show when minibatching helps a subgradient scheme with random feasibility updates., Comment: 31 pages

The Kaczmarz algorithm is a simple iterative scheme for solving consistent linear systems. At each step, the method projects the current iterate onto the solution space of a single constraint. Hence, it requires very low cost per iteration and storage, and it has a linear rate of convergence. Distributed implementations of Kaczmarz have become, in recent years, the de facto architectural choice for large-scale linear systems. Therefore, in this paper we develop a family of randomized block Kaczmarz algorithms that uses at each step a subset of the constraints and extrapolated stepsizes, and can be deployed on distributed computing units. Our approach is based on several new ideas and tools, including stochastic selection rule for the blocks of rows, stochastic conditioning of the linear system, and novel strategies for designing extrapolated stepsizes. We prove that randomized block Kaczmarz algorithm converges linearly in expectation, with a rate depending on the geometric properties of the matrix and its submatrices and on the size of the blocks. Our convergence analysis reveals that the algorithm is most effective when it is given a good sampling of the rows into well-conditioned blocks. Besides providing a general framework for the design and analysis of randomized block Kaczmarz methods, our results resolve an open problem in the literature related to the theoretical understanding of observed practical efficiency of extrapolated block Kaczmarz methods., Comment: 20 pages

In this paper we study the problem of model reduction of linear network systems. We aim at computing a reduced order stable approximation of the network with the same topology and optimal w.r.t. H2 norm error approximation. Our approach is based on time-domain moment matching framework, where we optimize over families of parameterized reduced order models matching a set of moments at arbitrary interpolation points. The parameterization of the low order models is in terms of the free parameters and of the interpolation points. For this family of parameterized models we formulate an optimization-based model reduction problem with the H2 norm of error approximation as objective function while the preservation of some structural and physical properties yields the constraints. This problem is nonconvex and we write it in terms of the Gramians of a minimal realization of the error system. We propose two solutions for this problem. The first solution assumes that the error system admits a block diagonal observability Gramian, allowing for a simple convex reformulation as semidefinite programming, but at the cost of some performance loss. We also derive sufficient conditions to guarantee block diagonalization of the Gramian. The second solution employs a gradient projection method for a smooth reformulation yielding (locally) optimal interpolation points and free parameters. The potential of the methods is illustrated on a power network., Comment: 11 pages

We propose a stochastic gradient framework for solving stochastic composite convex optimization problems with (possibly) infinite number of linear inclusion constraints that need to be satisfied almost surely. We use smoothing and homotopy techniques to handle constraints without the need for matrix-valued projections. We show for our stochastic gradient algorithm $\mathcal{O}(\log(k)/\sqrt{k})$ convergence rate for general convex objectives and $\mathcal{O}(\log(k)/k)$ convergence rate for restricted strongly convex objectives. These rates are known to be optimal up to logarithmic factors, even without constraints. We demonstrate the performance of our algorithm with numerical experiments on basis pursuit, a hard margin support vector machines and a portfolio optimization and show that our algorithm achieves state-of-the-art practical performance.

In this paper we compute families of reduced order models that match a prescribed set of moments of a highly dimensional linear time-invariant system. First, we fully parametrize the models in the interpolation points and in the free parameters, and then we fix the set of interpolation points and parametrize the models only in the free parameters. Based on these two parametrizations and using as objective function the H2-norm of the error approximation we derive non-convex optimization problems, i.e., we search for the optimal free parameters and even the interpolation points to determine the approximation model yielding the minimal H2-norm error. Further, we provide the necessary first-order optimality conditions for these optimization problems given explicitly in terms of the controllability and the observability Gramians of a minimal realization of the error system. Using the optimality conditions, we propose gradient type methods for solving the corresponding optimization problems, with mathematical guarantees on their convergence. We also derive convex SDP relaxations for these problems and analyze when the convex relaxations are exact. We illustrate numerically the efficiency of our results on several test examples., Comment: 25 pages

In this paper we consider large-scale smooth optimization problems with multiple linear coupled constraints. Due to the non-separability of the constraints, arbitrary random sketching would not be guaranteed to work. Thus, we first investigate necessary and sufficient conditions for the sketch sampling to have well-defined algorithms. Based on these sampling conditions we developed new sketch descent methods for solving general smooth linearly constrained problems, in particular, random sketch descent and accelerated random sketch descent methods. From our knowledge, this is the first convergence analysis of random sketch descent algorithms for optimization problems with multiple non-separable linear constraints. For the general case, when the objective function is smooth and non-convex, we prove for the non-accelerated variant sublinear rate in expectation for an appropriate optimality measure. In the smooth convex case, we derive for both algorithms, non-accelerated and accelerated random sketch descent, sublinear convergence rates in the expected values of the objective function. Additionally, if the objective function satisfies a strong convexity type condition, both algorithms converge linearly in expectation. In special cases, where complexity bounds are known for some particular sketching algorithms, such as coordinate descent methods for optimization problems with a single linear coupled constraint, our theory recovers the best-known bounds. We also show that when random sketch is sketching the coordinate directions randomly produces better results than the fixed selection rule. Finally, we present some numerical examples to illustrate the performances of our new algorithms., Comment: 28 pages

We introduce new global and local inexact oracle concepts for a wide class of convex functions in composite convex minimization. Such inexact oracles naturally come from primal-dual framework, barrier smoothing, inexact computations of gradients and Hessian, and many other situations. We also provide examples showing that the class of convex functions equipped with the newly inexact second-order oracles is larger than standard self-concordant as well as Lipschitz gradient function classes. Further, we investigate several properties of convex and/or self-concordant functions under the inexact second-order oracles which are useful for algorithm development. Next, we apply our theory to develop inexact proximal Newton-type schemes for minimizing general composite convex minimization problems equipped with such inexact oracles. Our theoretical results consist of new optimization algorithms, accompanied with global convergence guarantees to solve a wide class of composite convex optimization problems. When the first objective term is additionally self-concordant, we establish different local convergence results for our method. In particular, we prove that depending on the choice of accuracy levels of the inexact second-order oracles, we obtain different local convergence rates ranging from $R$-linear and $R$-superlinear to $R$-quadratic. In special cases, where convergence bounds are known, our theory recovers the best known rates. We also apply our settings to derive a new primal-dual method for composite convex minimization problems. Finally, we present some representative numerical examples to illustrate the benefit of our new algorithms., Comment: 28 pages, 6 figures, and 2 tables

Finding a point in the intersection of a collection of closed convex sets, that is the convex feasibility problem, represents the main modeling strategy for many computational problems. In this paper we analyze new stochastic reformulations of the convex feasibility problem in order to facilitate the development of new algorithmic schemes. We also analyze the conditioning problem parameters using certain (linear) regularity assumptions on the individual convex sets. Then, we introduce a general random projection algorithmic framework, which extends to the random settings many existing projection schemes, designed for the general convex feasibility problem. Our general random projection algorithm allows to project simultaneously on several sets, thus providing great flexibility in matching the implementation of the algorithm on the parallel architecture at hand. Based on the conditioning parameters, besides the asymptotic convergence results, we also derive explicit sublinear and linear convergence rates for this general algorithmic framework., Comment: 33 pages. First version published in March 2017