Two stochastic optimization algorithms for convex optimization with fixed point constraints

Nonsmooth convex optimization problems are solved over fixed point sets of nonexpansive mappings by using a distributed optimization technique. This is done for a networked system with an operator, who manages the system, and a finite number of users, by solving the problem of minimizing the sum of the operator’s and users’ nondifferentiable, convex objective functions over the intersection of the operator’s and users’ convex constraint sets in a real Hilbert space. We assume that each of their constraint sets can be expressed as the fixed point set of an implementable nonexpansive mapping. This setting allows us to discuss nonsmooth convex optimization problems in which the metric projection onto the constraint set cannot be calculated explicitly. We propose a parallel subgradient algorithm for solving the problem by using the operator’s attribution such that it can communicate with all users. The proposed algorithm does not use any proximity operators, in contrast to conventional parallel algorithms for nonsmooth convex optimization. We first study its convergence property for a constant step-size rule. The analysis indicates that the proposed algorithm with a small constant step size approximates a solution to the problem. We next consider the case of a diminishing step-size sequence and prove that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. We also give numerical examples to support the convergence analyses.

This paper proposes a decentralized optimization algorithm for the triple-hierarchical constrained convex optimization problem of minimizing a sum of strongly convex functions subject to a paramonotone variational inequality constraint over an intersection of fixed point sets of nonexpansive mappings. The existing algorithms for solving this problem are centralized optimization algorithms using all the information in the problem, and these algorithms are effective, but only under certain additional restrictions. The main contribution of this paper is to present a convergence analysis of the proposed algorithm in order to show that the proposed algorithm using incremental gradients with diminishing step-size sequences converges to the solution to the problem without any additional restrictions. Another contribution of this paper is the elucidation of the practical applications of hierarchical constrained optimization in the form of network resource allocation and optimal control problems. In particular, it is shown that the proposed algorithm can be applied to decentralized network resource allocation with a triple-hierarchical structure.

A viscosity method for a hierarchical fixed point approach to variational inequality problems is presented. This method is used to solve variational inequalities where the involving operators are complements of nonexpansive mappigs and the solutions are sought in the set of the fixed points of another nonexpansive mapping. Such variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings.

Random projection algorithm is an iterative gradient method with random projections. Such an algorithm is of interest for constrained optimization when the constraint set is not known in advance or the projection operation on the whole constraint set is computationally prohibitive. This paper presents a distributed random projection (DRP) algorithm for fully distributed constrained convex optimization problems that can be used by multiple agents connected over a time-varying network, where each agent has its own objective function and its own constrained set. With reasonable assumptions, we prove that the iterates of all agents converge to the same point in the optimal set almost surely. In addition, we consider a variant of the method that uses a mini-batch of consecutive random projections and establish its convergence in almost sure sense. Experiments on distributed support vector machines demonstrate fast convergence of the algorithm. It actually shows that the number of iteration required until convergence is much smaller than scanning over all training samples just once.

In this paper, we introduce two algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Furthermore, we prove that the proposed algorithms converge strongly to a solution of the minimization problem of finding x∗ ∈ Γ such that x∗= minx∈Γ x where Γ stands for the intersection set of the solution set of the equilibrium problem and the fixed points set of a nonexpansive mapping.

In 2006, Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, {\em SIAM J. Optim.}, 16(4) (2006), 1230-1241.] introduced an iterative algorithm for finding a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality in a real Hilbert space via combining two well-known methods: hybrid and extragradient. In this paper, motivated by Nadezhkina and Takahashi's hybrid-extragradient method we propose and analyze a relaxed hybrid-extragradient method for finding a solution of a constrained convex minimization problem, which is also a common element of the solution set of a variational inclusion and the fixed point set of a strictly pseudocontractive mapping in a real Hilbert space. We obtain a strong convergence theorem for three sequences generated by this algorithm. Based on this result, we also construct an iterative algorithm for finding a solution of the constrained convex minimization problem, which is also a common fixed point of two mappings taken from the more general class of strictly pseudocontractive mappings.

We consider a general multi-agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint, a global equality constraint, and a global constraint set. The objective function is defined by a sum of local objective functions, while the global constraint set is produced by the intersection of local constraint sets. In particular, we study two cases: one where the equality constraint is absent, and the other where the local constraint sets are identical. We devise two distributed primal-dual subgradient algorithms based on the characterization of the primal-dual optimal solutions as the saddle points of the Lagrangian and penalty functions. These algorithms can be implemented over networks with dynamically changing topologies but satisfying a standard connectivity property, and allow the agents to asymptotically agree on optimal solutions and optimal values of the optimization problem under the Slater's condition.

This paper presents a survey of the most recent literature developments in the field of computational guidance and control for space applications. In recent years, the space sector has changed significantly, shifting towards solutions requiring increased performance and autonomy. This change has also been accelerated by the technical and economic needs of the new space approach. However, traditional guidance and control techniques cannot always meet the increasing requirements of autonomy and performance. For this reason, computational guidance and control has become of increasing interest to the aerospace sector. In contrast to traditional guidance and control, computational guidance and control generates commands in real time by using on-board numerical computation. It represents, therefore, a fundamental step on the way to system autonomy and autonomous operations, with the potential to increase system capabilities and reduce operational costs. On-board computation introduces, however, a series of challenges, including the need to develop algorithms which are computationally efficient, reliable and robust. The literature has classified computational guidance and control techniques into two major categories, namely model-based and data-based. The model-based category encompasses all techniques in which the guidance and control laws are explicitly related to the system dynamic and kinematic models. On the contrary, data-based techniques infer guidance and control laws from a set of data, derived from the models, but do not explicitly use such models in the algorithm itself. Model-based guidance and control includes optimal control, model predictive control and convex optimisation. For the specific case of space attitude re-orientation manoeuvres, additional techniques are path planning and artificial potential functions. The general concept behind the Model Predictive Control (MPC) framework is to solve an optimal control problem repeatedly, for example whenever new measurements are available. The MPC implements only the first element of the returned solution, and the process is then repeated. By doing so, the system is better able to cope autonomously with model uncertainties, failures, or errors. For spacecraft with limited on-board capability, having to repeatedly solve an optimisation problem on-board could represent a challenge. In this case, explicit implementations of the MPC have been proposed, in which the guidance and control laws are precomputed offline and implemented on-board using look-up tables or simple functions. If the optimal control problem of the MPC is convex, or if it can be convexified, convex optimisation can be used directly. If the problem cannot be convexified, the solution can be found by sequentially solving a convex approximation of the original non-convex problem. Convex optimisation can be used either repeatedly in the context of an MPC framework, or as a single function call to generate guidance and control laws, e.g. at the beginning of an attitude re-orientation manoeuvre. Convex optimisation has received increased attention in recent years, also due to its suitability to on-board implementation. In fact, if the optimal control problem is convex, it can be solved reliably and efficiently on-board. Convex optimisation has been successfully applied to the space sector with the notable examples of the SpaceX rocket booster landing and of the Mars pinpoint landing demonstration aboard the Masten Xombie sounding rocket. Model-based guidance techniques, not based on optimisation, have been increasingly explored in recent years, as guidance strategies employed in other engineering fields, such as robotics, are being applied to spacecraft. Among these strategies, of particular importance are path planning and artificial potential field methods, due to their applicability to spacecraft autonomous attitude manoeuvres. Path planning algorithms have been applied for decades to ground vehicles and robotic manipulators. These algorithms consist of techniques to find the best route between two points of a graph representing the search space of a problem. For spacecraft re-orientation applications, the literature has proposed different attitude representation conventions and various cost functions associated to the graph search algorithm. Each approach comes with advantages and disadvantages, which will be analysed in the paper. Along with path planning, Artificial Potential Function (APF) techniques have been developed in the field of robotics, and are now also considered for space applications, to track or maintain a specific reference under a set of constraints. The main idea behind these methods is to synthetise the controller using a potential function. This function is specifically built to grant asymptotic stability to the system, according to Lyapunov’s second stability theorem. This formulation grants flexibility to accommodate system constraints, in a computationally light-weight and efficient manner. Data-based guidance and control laws are mainly based on Artificial Intelligence (AI) technologies, and generate guidance commands after training an algorithm with a large set of data. The interest of the space sector towards AI has accelerated in the last years due to the greater maturity of the technology. Some of the most powerful examples lie in the area of machine learning, like reinforcement learning, supervised learning and deep learning. These algorithms are used to extract patterns from large amounts of data derived from complex nonlinear dynamical and kinematical models. Examples of such data can be numerical solutions of an optimal control problem. The patterns extracted from the data allow the system to identify, in a short computational time, the critical parameters needed to relate the state and control variables. Creating a sufficiently large dataset for training the algorithm requires a considerable computational effort, though, and for certain methods optimality cannot be guaranteed due to the lack of theoretical support derived from the optimality conditions. Advancements made in the last years span several research directions and open a wide variety of opportunities and challenges, which need to be examined to guide further exploration. In this paper, both model-based and data-based techniques and applications are presented. The considered techniques include model predictive control, convex optimisation and machine learning. In addition, for the implementation of autonomous attitude manoeuvres, path planning methods and artificial potential function methods are analysed. These techniques have been applied, in the literature, to a range of different applications. The applications discussed in this paper include, in addition to attitude manoeuvres, powered descent landing guidance, low-thrust orbital transfers, proximity operations and station keeping.

Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the “length” of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these “sparse” optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/e) iterations to obtain an e-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/ 3 ) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/e) and O(1/ 3 ) iteration complexity bounds for obtaining an e-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.

The usual approach to developing and analyzing first-order methods for nonsmooth (stochastic or deterministic) convex optimization assumes that the objective function is uniformly Lipschitz continuous with parameter Mf. However, in many settings, the nondifferentiable convex function f is not uniformly Lipschitz continuous—for example, (i) the classical support vector machine problem, (ii) the problem of minimizing the maximum of convex quadratic functions, and even (iii) the univariate setting with [Formula: see text]. Herein, we develop a notion of “relative continuity” that is determined relative to a user-specified “reference function” h (that should be computationally tractable for algorithms), and we show that many nondifferentiable convex functions are relatively continuous with respect to a correspondingly fairly simple reference function h. We also similarly develop a notion of “relative stochastic continuity” for the stochastic setting. We analyze two standard algorithms—the (deterministic) mirror descent algorithm and the stochastic mirror descent algorithm—for solving optimization problems in these new settings, providing the first computational guarantees for instances where the objective function is not uniformly Lipschitz continuous. This paper is a companion paper for nondifferentiable convex optimization to the recent paper by Lu et al. [Lu H, Freund RM, Nesterov Y (2018) Relatively smooth convex optimization by first-order methods, and applications. SIAM J. Optim. 28(1): 333–354.], which developed analogous results for differentiable convex optimization.

In this paper, we consider constrained optimization problems subject to a convex set C. Stochastic gradient descent (SGD) is a simple and popular stochastic optimization algorithm that has been the workhorse of machine learning for many years. We show a new and surprising fact about SGD, in that depending on the constraint set C, one can operate SGD with much lower-dimensional stochastic gradients without affecting its performance. In particular, we design an optimization algorithm that operates with the lower-dimensional (compressed) stochastic gradients, and establish that with the right set of parameters it has the exact same dimension-free convergence guarantees as that of regular SGD for popular convex and nonconvex optimization settings. We also present two applications of these bounds, one for improving the empirical risk minimization bounds in differentially private nonconvex optimization, and other for reducing communication costs with distributed SGD. Additionally, we also show that this connection between constraint set structure and gradient compression also extends beyond SGD to the conditional gradient (Frank-Wolfe) method. The geometry of the constraint set, captured by its Gaussian width, plays an important role in all our results.

Projected subgradient based distributed convex optimization with transmission noises

In this paper, we study attractive points for a class of generalized nonexpansive mappings on star-shaped sets and establish strong convergence theorems of the Halpern iterative sequences generated by these mappings in a real Hilbert space. We modify Halpern’s iterations for finding an attractive point of a mapping T satisfying condition (E) on a star-shaped set C in a real Hilbert space H and provide an affirmative answer to an open problem posed by Akashi and Takahashi in a recent work of (Appl. Math. Comput. 219(4):2035-2040, 2012) for nonexpansive and nonspreading mappings. Furthermore, we study the approximation of common attractive points of generalized nonexpansive mappings and derive a strong convergence theorem by a new iteration scheme for these mappings. As applications of our results, we study multiple sets split monotone inclusion problems for inverse strongly monotone mappings, multiple sets split optimization problems, and multiple sets split feasibility problems. Our results contain many original results on multiple sets split feasibility problem in the literature. Our results also improve and generalize many well-known results in the current literature. MSC:47H10, 37C25.

We consider a general class of convex optimization problems over time-varying, multi-agent networks, that naturally arise in many application domains like energy systems and wireless networks. In particular, we focus on programs with separable objective functions, local (possibly different) constraint sets and a coupling inequality constraint expressed as the non-negativity of the sum of convex functions, each corresponding to one agent. We propose a novel distributed algorithm to deal with such problems based on a combination of dual decomposition and proximal minimization. Our approach is based on an iterative scheme that enables agents to reach consensus with respect to the dual variables, while preserving information privacy. Specifically, agents are not required to disclose information about their local objective and constraint functions, nor to assume knowledge of the coupling constraint. Our analysis can be thought of as a generalization of dual gradient/subgradient algorithms to a distributed set-up. We show convergence of the proposed algorithm to some optimal dual solution of the centralized problem counterpart, while the primal iterates generated by the algorithm converge to the set of optimal primal solutions. A numerical example demonstrating the efficacy of the proposed algorithm is also provided.

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.