Non-convex optimization problems arise from various areas in science and engineering. Although many numerical methods and theories have been developed for unconstrained non-convex problems, the parallel development for constrained non-convex problems remains limited. That restricts the practices of mathematical modeling and quantitative decision making in many disciplines. In this paper, an inexact proximal-point penalty method is proposed for constrained optimization problems where both the objective function and the constraint can be non-convex. The proposed method approximately solves a sequence of subproblems, each of which is formed by adding to the original objective function a proximal term and quadratic penalty terms associated to the constraint functions. Under a weak-convexity assumption, each subproblem is made strongly convex and can be solved effectively to a required accuracy by an optimal gradient-type method. The theoretical property of the proposed method is analyzed in two different cases. In the first case, the objective function is non-convex but the constraint functions are assumed to be convex, while in the second case, both the objective function and the constraint are non-convex. For both cases, we give the complexity results in terms of the number of function value and gradient evaluations to produce near-stationary points. Due to the different structures, different definitions of near-stationary points are given for the two cases. The complexity for producing a nearly @math -stationary point is @math for the first case while it becomes @math for the second case.
The augmented Lagrange method (ALM) @cite_32 @cite_20 @cite_18 @cite_35 is another common choice for constrained problems. Different from the exact or quadratic penalty method, ALM estimates the primal solution together with the dual solution. At each iteration, it updates the primal variable by minimizing the augmented Lagrange function and then performs a dual gradient ascent step to update the dual variable. The iteration complexity of ALM has been established for convex problems @cite_17 @cite_49 @cite_38 @cite_2 @cite_53 . For non-convex problems, most of the existing studies on ALM only show its asymptotic convergence or local convergence rate @cite_41 @cite_61 @cite_19 @cite_30 @cite_64 @cite_82 . The computational complexities of ALM for finding an @math -stationary point (under various notions of stationarity) are obtained only for linearly constrained problems @cite_39 @cite_12 @cite_63 @cite_4 . One exception is @cite_29 where they essentially assume that the smallest singular value of the Jacobian matrix of the constraint functions is uniformly bounded away from zero at all feasible points. In this paper, we do not require that assumption but, instead, need an initial nearly feasible solution when the constraints are non-convex while @cite_29 does not need.
@cite_30 @cite_35 @cite_64 @cite_41 @cite_29 @cite_2 @cite_20 @cite_38 @cite_18 @cite_4 @cite_39 @cite_49 @cite_17 @cite_32 @cite_19 @cite_12 @cite_82 @cite_61 @cite_53 @cite_63
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Augmented Lagrangian method (ALM) has been popularly used for solving constrained optimization problems. Practically, subproblems for updating primal variables in the framework of ALM usually can only be solved inexactly. The convergence and local convergence speed of ALM have been extensively studied. However, the global convergence rate of the inexact ALM is still open for problems with nonlinear inequality constraints. In this paper, we work on general convex programs with both equality and inequality constraints. For these problems, we establish the global convergence rate of the inexact ALM and estimate its iteration complexity in terms of the number of gradient evaluations to produce a primal and or primal-dual solution with a specified accuracy. We first establish an ergodic convergence rate result of the inexact ALM that uses constant penalty parameters or geometrically increasing penalty parameters. Based on the convergence rate result, we then apply Nesterov’s optimal first-order method on each primal subproblem and estimate the iteration complexity of the inexact ALM. We show that if the objective is convex, then (O( ^ -1 ) ) gradient evaluations are sufficient to guarantee a primal ( )-solution in terms of both primal objective and feasibility violation. If the objective is strongly convex, the result can be improved to (O( ^ - 1 2 | |) ). To produce a primal-dual ( )-solution, more gradient evaluations are needed for convex case, and the number is (O( ^ - 4 3 ) ), while for strongly convex case, the number is still (O( ^ - 1 2 | |) ). Finally, we establish a nonergodic convergence rate result of the inexact ALM that uses geometrically increasing penalty parameters. This result is established only for the primal problem. We show that the nonergodic iteration complexity result is in the same order as that for the ergodic result. Numerical experiments on quadratically constrained quadratic programming are conducted to compare the performance of the inexact ALM with different settings. 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. First-order methods have been popularly used for solving large-scale problems. However, many existing works only consider unconstrained problems or those with simple constraint. In this paper, we develop two first-order methods for constrained convex programs, for which the constraint set is represented by affine equations and smooth nonlinear inequalities. Both methods are based on the classic augmented Lagrangian function. They update the multipliers in the same way as the augmented Lagrangian method (ALM) but employ different primal variable updates. The first method, at each iteration, performs a single proximal gradient step to the primal variable, and the second method is a block update version of the first one. For the first method, we establish its global iterate convergence as well as global sublinear and local linear convergence, and for the second method, we show a global sublinear convergence result in expectation. Numerical experiments are carried out on the basis pursuit denoising and a convex quadratically constrained quadratic program to show the empirical performance of the proposed methods. Their numerical behaviors closely match the established theoretical results. In this paper, we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the diagonal quadratic approximation method DQAM of Mulvey and Ruszczynski [A diagonal quadratic approximation method for large scale linear programs, Oper. Res. Lett. 12 1992, pp. 205–215] and the parallel coordinate descent method PCDM of Richtarik and Takac [Parallel coordinate descent methods for big data optimization. Technical report, November 2012. arXiv:1212.0873]. We show that the two methods are equivalent for feasibility problems up to the selection of a step-size parameter. Furthermore, we prove an improved complexity bound for PCDM under strong convexity, and show that this bound is at least 8L′ Lω−12 times better than the best known bound for DQAM, where ω is the degree of partial separability and L’ and L are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian.