In this paper, we consider the convergence of an abstract inexact nonconvex and nonsmooth algorithm. We promise a pseudo sufficient descent condition and a pseudo relative error condition, which are both related to an auxiliary sequence, for the algorithm; and a continuity condition is assumed to hold. In fact, a lot of classical inexact nonconvex and nonsmooth algorithms allow these three conditions. Under a special kind of summable assumption on the auxiliary sequence, we prove the sequence generated by the general algorithm converges to a critical point of the objective function if being assumed Kurdyka- Lojasiewicz property. The core of the proofs lies in building a new Lyapunov function, whose successive difference provides a bound for the successive difference of the points generated by the algorithm. And then, we apply our findings to several classical nonconvex iterative algorithms and derive the corresponding convergence results
Recently, the convergence analysis in nonconvex optimization has paid increasing attention to using the semi-algebraic property in proofs. In paper @cite_25 , the authors proved the convergence of proximal algorithm minimizing the semi-algebraic functions. In @cite_25 , the rates for the iteration converging to a critical point were exploited. An alternating proximal algorithm was considered in @cite_19 , and the convergence was proved under semi-algebraic assumption on the objective function. Latter, a proximal linearized alternating minimization algorithm was proposed and studied in @cite_24 . A convergence framework was given in @cite_14 , which contains various nonconvex algorithms. In @cite_5 , the authors modified the framework for analyzing splitting methods with variable metric, and proved the general convergence rates. The nonconvex ADMM was studied under semi-algebraic assumption by @cite_26 @cite_0 . And latter paper @cite_30 proposed the nonconvex primal-dual algorithm and proved the convergence. The semi-algebraic analysis convergence method was applied to analyzing the convergence of the reweighted algorithm by @cite_4 . And the extension to the reweighted nuclear norm version was developed in @cite_17 . Recently, the DC algorithm has also employed the semi-algebraic property in the convergence analysis @cite_35 .
@cite_30 @cite_35 @cite_14 @cite_26 @cite_4 @cite_24 @cite_19 @cite_0 @cite_5 @cite_25 @cite_17
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We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x,y)=f(x)+Q(x,y)+g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to “metrically regular” problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to @math and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with “regular” intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l1 algorithm for compressive sensing and an application to rank reduction problems. In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, ( (x_0, ,x_p,y) ), subject to coupled linear equality constraints. Our ADMM updates each of the primal variables (x_0, ,x_p,y ), followed by updating the dual variable. We separate the variable y from (x_i )’s as it has a special role in our analysis. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, ( _q ) quasi-norm, Schatten-q quasi-norm ( (0<q<1 )), minimax concave penalty (MCP), and smoothly clipped absolute deviation penalty. It also allows nonconvex constraints such as compact manifolds (e.g., spherical, Stiefel, and Grassman manifolds) and linear complementarity constraints. Also, the (x_0 )-block can be almost any lower semi-continuous function. By applying our analysis, we show, for the first time, that several ADMM algorithms applied to solve nonconvex models in statistical learning, optimization on manifold, and matrix decomposition are guaranteed to converge. Our results provide sufficient conditions for ADMM to converge on (convex or nonconvex) monotropic programs with three or more blocks, as they are special cases of our model. ADMM has been regarded as a variant to the augmented Lagrangian method (ALM). We present a simple example to illustrate how ADMM converges but ALM diverges with bounded penalty parameter ( ). Indicated by this example and other analysis in this paper, ADMM might be a better choice than ALM for some nonconvex nonsmooth problems, because ADMM is not only easier to implement, it is also more likely to converge for the concerned scenarios. Nonconvex optimization arises in many areas of computational science and engineering. However, most nonconvex optimization algorithms are only known to have local convergence or subsequence convergence properties. In this paper, we propose an algorithm for nonconvex optimization and establish its global convergence (of the whole sequence) to a critical point. In addition, we give its asymptotic convergence rate and numerically demonstrate its efficiency. In our algorithm, the variables of the underlying problem are either treated as one block or multiple disjoint blocks. It is assumed that each non-differentiable component of the objective function, or each constraint, applies only to one block of variables. The differentiable components of the objective function, however, can involve multiple blocks of variables together. Our algorithm updates one block of variables at a time by minimizing a certain prox-linear surrogate, along with an extrapolation to accelerate its convergence. The order of update can be either deterministically cyclic or randomly shuffled for each cycle. In fact, our convergence analysis only needs that each block be updated at least once in every fixed number of iterations. We show its global convergence (of the whole sequence) to a critical point under fairly loose conditions including, in particular, the Kurdyka–Łojasiewicz condition, which is satisfied by a broad class of nonconvex nonsmooth applications. These results, of course, remain valid when the underlying problem is convex. We apply our convergence results to the coordinate descent iteration for non-convex regularized linear regression, as well as a modified rank-one residue iteration for nonnegative matrix factorization. We show that both applications have global convergence. Numerically, we tested our algorithm on nonnegative matrix and tensor factorization problems, where random shuffling clearly improves the chance to avoid low-quality local solutions. In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.