Algorithms, visualization, software, and applications nonconvex optimization and its applications on free. Why should nonconvexity be a problem in optimization. Interiorpoint methods these methods are best suited for convex optimization, but perform remarkably well on nonconvex optimization. Convex and nonconvex optimization methods for machine learning. Using the convex envelope of multilinear functions as our.
Siam journal on optimization siam society for industrial. Berkeley nonconvex problems are 2 nonconvex optimization problem with simple constraints question. However, the convergence guarantees of adaptive gradient methods for nonconvex optimization have not been suciently studied. Simple convex example illustration of failure and success methods suitable for nonsmooth functions gradient sampling quasinewton methods a.
Our analysis shows that under this scenario such methods. Combining dca dc algorithms and interior point techniques. All journal articles featured in optimization methods and software vol 35 issue 2. An accessible analysis of stochastic convex optimization. Sahinidis, global optimization of nonconvex problems with convextransformable intermediates, journal of global optimization. Nonconvex optimization of desirability functions request pdf. Algorithms, visualization, software, and applications nonconvex optimization and its applications on free shipping on qualified orders. An approach based on the kurdykalojasiewicz inequality. What are some recent advances in nonconvex optimization. This dissertation is concerned with modeling fundamental and challenging machine learning tasks as convexnonconvex optimization.
We propose an algorithm for solving nonsmooth, nonconvex, constrained optimization problems as well as a new set of visualization tools for comparing the performance of optimization algorithms. What is the difference between convex and nonconvex optimization. The global solver in the lindo application programming interface lindo api finds guaranteed global optima to nonconvex, nonlinear and integer mathematical models using the branch and boundrelax approach. On the global convergence of the bfgs method for nonconvex. The new algorithm performs explicit matrix modifications adaptively, mimicing the implicit modifications used by trustregion methods. Global convergence of splitting methods for nonconvex composite optimization guoyin li ting kei pong y november 30, 2014 abstract we consider the problem of minimizing the sum of a smooth function hwith a bounded hessian, and a nonsmooth function. This research mainly focuses on designing algorithms for distributed nonconvex optimization problems under di erent network topologies. Visual computing center kaust, thuwal saudi arabia samuel.
On the convergence of adaptive gradient methods for. Global solution of nonconvex quadratically constrained. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a nonconvex function. The numerical results on the cuter problems demonstrate the effectiveness of this approach in the context of a linesearch method for largescale unconstrained nonconvex optimization. Theory and applications yu wang xian jiaotong, wotao yin ucla, jinshan zeng jiangxi normal. We propose a trustregion type method for general nonsmooth nonconvex optimization problems with emphasis on nonsmooth composite programs where the objective function is a summation of a. Mosek is designed to solve largescale convex optimization problems including linear programming, socp, semidefinite programming, and other convex problems. We provide a brief description of the algorithms used by the software, describe the types of problems that can be currently solved and summarize our recent computational experience. A bfgssqp method for nonsmooth, nonconvex, constrained. In this paper, we have developed a new algorithm for solving nonconvex large scale problems. A new branchandbound algorithm for standard quadratic programming problems. Algorithms for solving a class of nonconvex optimization problems. Referenced in 31 articles memory bundle method for largescale nonsmooth optimization many practical optimization problems involve nonsmooth that.
Strekalovsky russianacademyofsciences, siberianbranch, instituteforsystemdynamicsandcontroltheory. The combined homotopy methods for optimization problem in non. The chapter adapts the algorithms to solve the problems. In this work, we provide a new analysis of such methods applied to nonconvex stochastic optimization problems, characterizing the effect of increasing minibatch size. Nonconvex optimization for communication systems princeton. Stochastic proximal quasinewton methods for nonconvex. Pdf stochastic proximal quasinewton methods for non. Distributed nonconvex optimization problem has found a wide range of applications in several areas, including dataintensive optimization 65, 146. Issues in nonconvex optimization mit opencourseware. The alternating direction method of multipliers adm or admm appeared in glowinski and marroco75, gabay and mercier76. The methods of subgradients are very easy to program and are actually the only ones which enables to. Thus, there is a fundamental gap in our understanding of stochastic methods for nonsmooth nonconvex problems.
A stochastic semismooth newton method for nonsmooth. This paper presents a fully asynchronous and distributed approach for tackling optimization problems in which both the objective function and the constraints may be nonconvex. Comparison of nonsmooth and nonconvex optimization methods can be found in 1, and details on computational contact mechanics is presented in 19. The test bed includes convex and nonconvex problems, smooth as well as nonsmooth. Moreover, we further develop the branchingandsampling and branchinganddiscarding approaches to improve the objective value for 01 programs. While previously, the focus was on convex relaxation methods, now the. This article addresses the generation of strong polyhedral relaxations for nonconvex, quadratically constrained quadratic programs qcqps. Rankconstrained programming for nonconvex optimization.
Nonconvex optimization methods for sparse and lowrank. Abstract adaptive gradient methods are workhorses in deep learning. Primaldual activeset methods for convex quadratic optimization pypdas. The 4th conference on optimization methods and software, december 1620, 2017, havana, cuba. Stochastic quasinewton methods for nonconvex stochastic. Nonsmooth, nonconvex optimization introduction nonsmooth, nonconvex optimization example methods suitable for nonsmooth functions failure of steepest descent. Stochastic first and zerothorder methods for nonconvex stochastic programming. Adaptivity of stochastic gradient methods for nonconvex.
Gradientbased algorithm for nonsmooth optimization. Proximal point methods and nonconvex optimization springerlink. Simpler example gradient sampling quasinewton methods some di. Nonconvex optimization has either a nonconvex domain or the objective function. However, since it does not satisfy the triangle inequality nor symmetry, 1the use of bregman distance in optimization within various contexts is well spread. You can use some methods to identity a convex or not by. The attained rates improve over both sgd and graddescent, a bene. A disadvantage of the sdp semidefinite programming relaxation method for quadratic andor combinatorial optimization problems lies in its expensive computational cost. Proximal alternating minimization and projection methods for. Minibatch stochastic approximation methods for nonconvex stochastic composite optimization. A matrixfree linesearch algorithm for nonconvex optimization. The global solver in the lindo api optimization methods. Audience the book is of interest to both researchers in operations research, systems engineering, and optimization methods, as well as applications specialists concerned with the solution of large scale discrete andor nonconvex optimization problems in a broad range of engineering and technological fields.
Convergence analysis of proximal gradient with momentum. Numerical studies show three advantages of the proposed methods. Given the ubiquity of nonconvex models in machine learning. In this paper we study stochastic quasinewton methods for nonconvex stochastic optimization, where we assume that noisy information about the gradients of the objective function is available via a. Napsu karmitsa nonsmooth optimization nso software. We show that the algorithm is well suited for solving very largescale nonconvex problems whenever hessianvector products are available.
Distributed nonconvex optimization problem has found a wide range of applications in several areas, including dataintensive optimization. Proximal bundle method for nonsmooth possibly nonconvex multiobjective minimization by m. A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In this work, we present a globalized stochastic semismooth newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective. Therefore, there is a huge gap between existing online convex optimization guarantees for adaptive gradient methods and the empirical successes of adaptive gradient methods in nonconvex optimization. Many classes of convex optimization problems admit polynomialtime algorithms, whereas mathematical optimization is in general nphard. Simple convex example illustration of failure and success methods suitable for nonsmooth functions gradient sampling quasinewton methods. Global convergence of splitting methods for nonconvex. A general system for heuristic minimization of convex. Toward designing convergent deep operator splitting.
Smoothing nonlinear conjugate gradient method for image. A variety of nonconvex optimization techniques are showcased. The combined homotopy methods for optimization problem in nonconvex. The optimization algorithm for socp used together with yalmip in this paper is a stateoftheart primaldual interiorpoint algorithm implemented in the software mosek. The book covers both the theory and the numerical methods used in nso and provide an overview of di. Nonconvex optimization methods for largescale statistical. We assume that the latter function is a composition of. Visual computing center kaust, thuwal saudi arabia peter. Proximal stochastic methods for nonsmooth nonconvex finite. Adaptivity of stochastic gradient methods for nonconvex optimization samuel horvath. Adaptivity of stochastic gradient methods for nonconvex optimization. Stochastic variance reduction for nonconvex optimization. A general system for heuristic minimization of convex functions. Constructivism dc, 1, and allowing calculus to be used also in software, for example, which is.
In this work, we present a globalized stochastic semismooth newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the. The example also shows how a modeling system can vastly simplify the process of converting a convex optimization. Python software for a primaldual activeset method for solving general convex quadratic optimization. Modern methods for nonconvex optimization problems alexander s.
Convex optimization has provided both a powerful tool and an intriguing mentality to the analysis and design of communication systems over the last few years. On the convergence of adaptive gradient methods for nonconvex optimization. Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization xiaojun chen. Yes, nonconvex optimization is at least nphard can encode most problems as nonconvex optimization problems example. Fast stochastic methods for nonsmooth nonconvex optimization anonymous authors af. The new algorithm performs explicit matrix modifications. In this paper, we have developed a new algorithm for solving nonconvex largescale problems. Chanceconstrained optimization for nonconvex programs using. Proximal alternating minimization and projection methods for nonconvex problems.
This paper proposes a socp secondordercone programming relaxation method, which strengthens the liftandproject lp linear programming relaxation method by adding convex quadratic valid inequalities for the positive. Fast incremental method for nonconvex optimization sashank j. The 4th conference on optimization methods and software. Toward designing convergent deep operator splitting methods for taskspeci. Advances in multimedia, software engineering and computing vol. Along with many derivativefree algorithms, many software implementations have also appeared. Buy bayesian heuristic approach to discrete and global optimization. Smoothing methods for nonsmooth, nonconvex minimization. Toward designing convergent deep operator splitting methods. The opensource software for solving qcqps is published at.
Nonconvex optimization is now ubiquitous in machine learning. Methods and software 15 anisms for switching to a feasibility restoration if the step size becomes too small. Faster stochastic alternating direction method of multipliers for nonconvex optimization feihu huang1 songcan chen2 3 heng huang1 4 abstract in this paper, we propose a faster stochastic alternating direction method of multipliers admm for nonconvex optimization. Based on this definition, we can construct a smoothing method. Bayesian heuristic approach to discrete and global. A general purpose global optimization software package.
Related work a concise survey of incremental gradient methods is 5. These methods have higher computational requirements at each iteration much more computations and more memory per iteration, but convergence rates of these methods are usually locally quadratic. The branchandreduce optimization navigator baron is a computational system for facilitating the solution of nonconvex optimization problems to global optimality. In order to bridge this gap, there are a few recent attempts to prove the nonconvex optimization guarantees for adaptive gradient methods. Stochastic proximal quasinewton methods for nonconvex composite optimization. This paper proposes a socp secondordercone programming relaxation method, which strengthens the liftandproject lp linear programming relaxation method. The example also shows how a modeling system can vastly simplify the process of converting a convex optimization problem into standard form. Sahinidis, exploiting integrality in the global optimization of mixedinteger nonlinear programming problems in baron, optimization methods and software, 33, 540562, 2018. Its important to note that this is really not the best way to. Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Multiobjective proximal bundle method for nonconvex nonsmooth optimization. While previously, the focus was on convex relaxation methods, now the emphasis is on being able to solve nonconvex problems directly. Inexact proximal stochastic secondorder methods for nonconvex composite optimization.
Solving nonconvex optimal control problems by convex optimization. Proximal stochastic methods for nonsmooth nonconvex. Newton methods for nonconvex composite optimization, optimization methods and software, doi. Nonsmooth, nonconvex optimization introduction nonsmooth, nonconvex optimization a simple nonconvex example failure of gradient descent in nonsmooth case armijowolfe line search failure of gradient method. A general system for heuristic minimization of convex functions over nonconvex sets. Request pdf nonconvex optimization of desirability functions desirability functions dfs are commonly used in optimization of design parameters with multiple quality characteristic to obtain. First, sequential optimization obtains an optimal solution much faster than the stateoftheart software. Python software for a primaldual activeset method for solving general convex quadratic optimization problems. Dongruo zhouy yiqi tangz ziyan yangx yuan cao quanquan guk. A general system for heuristic minimization of convex functions over. Interiorpoint methods these methods are best suited for convex optimization, but perform remarkably well on nonconvex optimization as well.