In this work, we shed light on the so-called Kaczmarz method for solving Linear System (LS) and Linear Feasibility (LF) problems from a optimization point of view. We introduce well-known optimization approaches such as Lagrangian penalty and Augmented Lagrangian in the Randomized Kaczmarz (RK) method. In doing so, we propose two variants of the RK method namely the Randomized Penalty Kacmarz (RPK) method and Randomized Augmented Kacmarz (RAK) method. We carry out convergence analysis of the proposed methods and obtain linear convergence results.
We develop two fundamental stochastic sketching techniques; Penalty Sketching (PS) and Augmented Lagrangian Sketching (ALS) for solving consistent linear systems. The proposed PS and ALS techniques extend and generalize the scope of Sketch & Project (SP) method by introducing Lagrangian penalty sketches. In doing so, we recover SP methods as special cases and furthermore develop a family of new stochastic iterative methods. By varying sketch parameters in the proposed PS method, we recover novel stochastic methods such as Penalty Newton Descent, Penalty Kaczmarz, Penalty Stochastic Descent, Pe…
We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. In doing so, we obtain a family of algorithms that generalize and extend both techniques. We prove several convergence results, and our computational experiments show our algorithms often outperform the original methods.
The Sampling Kaczmarz Motzkin (SKM) algorithm is a generalized method for solving large scale linear systems of inequalities. Having its root in the relaxation method of Agmon, Schoenberg, and Motzkin and the randomized Kaczmarz method, SKM outperforms the state of the art methods in solving large-scale Linear Feasibility (LF) problems. Motivated by SKM's success, in this work, we propose an Accelerated Sampling Kaczmarz Motzkin (ASKM) algorithm which achieves better convergence compared to the standard SKM algorithm on ill conditioned problems. We provide a thorough convergence analysis for t…
The randomized version of the Kaczmarz method for the solution of linear systems is known to converge linearly in expectation. In this work we extend this result and show that the recently proposed Randomized Sparse Kaczmarz method for recovery of sparse solutions, as well as many variants, also converges linearly in expectation. The result is achieved in the framework of split feasibility problems and their solution by randomized Bregman projections with respect to strongly convex functions. To obtain the expected convergence rates we prove extensions of error bounds for projections. The conv…
Randomized Kaczmarz (RK), Motzkin Method (MM) and Sampling Kaczmarz Motzkin (SKM) algorithms are commonly used iterative techniques for solving a system of linear inequalities (i.e., $Ax \leq b$). As linear systems of equations represent a modeling paradigm for solving many optimization problems, these randomized and iterative techniques are gaining popularity among researchers in different domains. In this work, we propose a Generalized Sampling Kaczmarz Motzkin (GSKM) method that unifies the iterative methods into a single framework. In addition to the general framework, we propose a Nestero…
The recently proposed Sampling Kaczmarz Motzkin (SKM) algorithm performs well in comparison with the state-of-the-art methods in solving large-scale Linear Feasibility (LF) problems. To explore the concept of momentum in the context of solving LF problems, in this work, we propose a momentum induced algorithm called Momentum Sampling Kaczmarz Motzkin (MSKM). The MSKM algorithm is developed by integrating the heavy ball momentum to the SKM algorithm. We provide a rigorous convergence analysis of the proposed MSKM algorithm from which we obtain convergence results of several Kaczmarz type method…
We develop two greedy sampling rules for the Sketch & Project method for solving linear feasibility problems. The proposed greedy sampling rules generalize the existing max-distance sampling rule and uniform sampling rule and generate faster variants of Sketch & Project methods. We also introduce greedy capped sampling rules that improve the existing capped sampling rules. Moreover, we incorporate the so-called heavy ball momentum technique to the proposed greedy Sketch & Project method. By varying the parameters such as sampling rules, sketching vectors; we recover several well-known algorith…
The randomized projection (RP) method is a simple iterative scheme for solving linear feasibility problems and has recently gained popularity due to its speed and low memory requirement. This paper develops an accelerated variant of the standard RP method by using two ingredients: the greedy probability criterion and the average block approach, and obtains a greedy randomized average block projection (GRABP) method for solving large-scale systems of linear inequalities. We prove that this method converges linearly in expectation under different choices of extrapolated stepsizes. Numerical expe…
Feasibility pumps are highly effective primal heuristics for mixed-integer linear and nonlinear optimization. However, despite their success in practice there are only few works considering their theoretical properties. We show that feasibility pumps can be seen as alternating direction methods applied to special reformulations of the original problem, inheriting the convergence theory of these methods. Moreover, we propose a novel penalty framework that encompasses this alternating direction method, which allows us to refrain from random perturbations that are applied in standard versions of …
A Two-Phase Adaptive Balanced Penalty Method for Controllable Pareto Front Learning under Split Feasibility Conditions
Faculty of Applied Mathematics and Informatics, Hanoi University of Science and Technology（应用数学与信息技术学院，河内科学技术大学）；College of Engineering and Computer Science, VinUniversity（工程与计算机科学学院，Vin大学）
Transforming into an exact penalty function model with convex compact constraints yields efficient infeasible approaches for optimization problems with orthogonality constraints. For smooth and $\ell_{2,1}$-norm regularized cases, these infeasible approaches adopt simple and orthonormalization-free updating scheme and show their high efficiency in the test examples. However, to avoid orthonormalization while enforcing the feasibility of the final solution, these infeasible approaches introduce a quadratic penalty term, where an inappropriate penalty parameter can lead to numerical inefficiency…
We propose a decentralized penalty method for general convex constrained multi-agent optimization problems. Each auxiliary penalized problem is solved approximately with a special parallel descent splitting method. The method can be implemented in a computational network where each agent sends information only to the nearest neighbours. Convergence of the method is established under rather weak assumptions. We also describe a specialization of the proposed approach to the feasibility problem.
We propose an implicit iterative algorithm for an exact penalty method arising from inequality constrained optimization problems. A rapidly convergent fixed point method is developed for a regularized penalty functional. The applicability and feasibility of the proposed method is demonstrated using large scale inequality constrained problems.
In this paper, we consider a stochastic Model Predictive Control able to account for effects of additive stochastic disturbance with unbounded support, and requiring no restrictive assumption on either independence nor Gaussianity. We revisit the rather classical approach based on penalty functions, with the aim of designing a control scheme that meets some given probabilistic specifications. The main difference with previous approaches is that we do not recur to the notion of probabilistic recursive feasibility, and hence we do not consider separately the unfeasible case. In particular, two p…
Simultaneous perturbation stochastic approximation (SPSA) is widely used in stochastic optimization due to its high efficiency, asymptotic stability, and reduced number of required loss function measurements. However, the standard SPSA algorithm needs to be modified to deal with constrained problems. In recent years, sequential quadratic programming (SQP)-based projection ideas and penalty ideas have been analyzed. Both ideas have convergence results and a potentially wide range of applications, but with some limitations in practical consideration, such as computation time, complexity, and fea…
This paper focuses on the design of sequential quadratic optimization (commonly known as SQP) methods for solving large-scale nonlinear optimization problems. The most computationally demanding aspect of such an approach is the computation of the search direction during each iteration, for which we consider the use of matrix-free methods. In particular, we develop a method that requires an inexact solve of a single QP subproblem to establish the convergence of the overall SQP method. It is known that SQP methods can be plagued by poor behavior of the global convergence mechanism. To confront t…
A penalty-type method for relaxed inverse optimal control problems
University of Siegen（锡根大学）；Higher Training College, University of Yaoundé I（雅温得第一大学高等培训学院）；AIMS-Cameroon Research Center（AIMS喀麦隆研究中心）；University of Marburg（马尔堡大学）
We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty lies in finding the projection of a point in the intersection of many sets. Existing approaches yield an infeasible point with an iteration-complexity of $O(1/\varepsilon^2)$ for nonsmooth problems with no guarantees on the in-feasibility. By reformulating the problem through exact penalty functions, we derive first-order algorithms which not only guarantee…
MoSSP: A Momentum-Based Single-Loop Stochastic Penalty Method for Nonconvex Constrained DC-Regularized Optimization
School of Mathematical Sciences, Beihang University, Beijing 100191, China（北京航空航天大学数学学院）；School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China（中山大学计算机科学与工程学院）
Optimal control problems with constraints ensuring safety and convergence to desired states can be mapped onto a sequence of real time optimization problems through the use of Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). One of the main challenges in these approaches is ensuring the feasibility of the resulting quadratic programs (QPs) if the system is affine in controls. The recently proposed penalty method has the potential to improve the existence of feasible solutions to such problems. In this paper, we further improve the feasibility robustness (i.e., feasibilit…
Euclidean distance matrix optimization with ordinal constraints (EDMOC) has found important applications in sensor network localization and molecular conformation. It can also be viewed as a matrix formulation of multidimensional scaling, which is to embed n points in a $r$-dimensional space such that the resulting distances follow the ordinal constraints. The ordinal constraints, though proved to be quite useful, may result in only zero solution when too many are added, leaving the feasibility of EDMOC as a question. In this paper, we first study the feasibility of EDMOC systematically. We sh…
We study the numerical reconstruction problem in acousto-electric tomography of recovering the conductivity distribution in a bounded domain from interior power density data. We propose a numerical method for recovering discontinuous conductivity distributions, by reformulating it as an optimization problem with $ L^1 $ fitting and total variation penalty subject to PDE constraints. We establish continuity and differentiability results for the forward map, the well-posedness of the optimization problem, and present an easy-to-implement and robust numerical method based on successive linearizat…
In this paper, we propose first-order feasible methods for difference-of-convex (DC) programs with smooth inequality and simple geometric constraints. Our strategy for maintaining feasibility of the iterates is based on a "retraction" idea adapted from the literature of manifold optimization. When the constraints are convex, we establish the global subsequential convergence of the sequence generated by our algorithm under strict feasibility condition, and analyze its convergence rate when the objective is in addition convex according to the Kurdyka-Lojasiewicz (KL) exponent of the extended obj…
Argonne National Laboratory（阿贡国家实验室）；The University of Chicago（芝加哥大学）；University of California, Berkeley（加州大学伯克利分校）；International Computer Science Institute（国际计算机科学研究所）
We study nonlinear optimization problems with a stochastic objective and deterministic equality and inequality constraints, which emerge in numerous applications including finance, manufacturing, power systems and, recently, deep neural networks. We propose an active-set stochastic sequential quadratic programming (StoSQP) algorithm that utilizes a differentiable exact augmented Lagrangian as the merit function. The algorithm adaptively selects the penalty parameters of the augmented Lagrangian and performs a stochastic line search to decide the stepsize. The global convergence is established:…
To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is particularly huge in some application areas such as materials computation. In this paper, we propose a proximal linearized augmented Lagrangian algorithm (PLAM) for solving optimization problems with orthogonality constraints. Unlike the classical augmented Lagrangian methods, in our algorithm, the prime variables are updated by minimizing a proximal linear…
We propose a multi-swarm approach to approximate the Pareto front of general multi-objective optimization problems that is based on the Consensus-based Optimization method (CBO). The algorithm is motivated step by step beginning with a simple extension of CBO based on fixed scalarization weights. To overcome the issue of choosing the weights we propose an adaptive weight strategy in the second modelling step. The modelling process is concluded with the incorporation of a penalty strategy that avoids clusters along the Pareto front and a diffusion term that prevents collapsing swarms. Altogethe…
The work of Wachter and Biegler suggests that infeasible-start interior point methods (IPMs) developed for linear programming cannot be adapted to nonlinear optimization without significant modification, i.e., using a two-phase or penalty method. We propose an IPM that, by careful initialization and updates of the slack variables, is guaranteed to find a first-order certificate of local infeasibility, local optimality or unboundedness of the (shifted) feasible region. Our proposed algorithm differs from other IPM methods for nonconvex programming because we reduce primal feasibility at the sam…
Compressed Traffic Assignment with the Augmented Lagrangian Method
Fulton School of Engineering, Arizona State University（亚利桑那州立大学富尔顿工程学院）；School of Sustainable Engineering and the Built Environment, ASU（亚利桑那州立大学可持续工程与建成环境学院）；Norfolk Southern Corporation（诺福克南方公司）；School of Computing and Augmented Intelligence, ASU（亚利桑那州立大学计算与增强智能学院）
