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.
On solving a problem with a non-convex objective and linear constraint, @cite_25 has developed a quadratic-penalty accelerated inexact proximal point method. That method can generate an @math -stationary point in the sense of with a complexity of @math . Our method is similar to that in @cite_25 by utilizing the techniques from both the proximal point method and the quadratic penalty method. Although we make a little stronger assumption than @cite_25 by requiring the boundedness of @math , our method and analysis apply to the problems with non-convex objectives and convex non-convex nonlinear constraint functions. When the constraints are convex (but possibly nonlinear), our method can find a nearly @math -stationary point with a complexity of @math that is a nearly @math improvement over the complexity in @cite_25 .
@cite_25
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This paper analyzes the iteration-complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs. More specifically, the objective function is of the form @math where @math is a differentiable function whose gradient is Lipschitz continuous and @math is a closed convex function with bounded domain. The method, basically, consists of applying an accelerated inexact proximal point method for solving approximately a sequence of quadratic penalized subproblems associated to the linearly constrained problem. Each subproblem of the proximal point method is in turn approximately solved by an accelerated composite gradient method. It is shown that the proposed scheme generates a @math -approximate stationary point in at most @math . Finally, numerical results showing the efficiency of the proposed method are also given. We develop two new proximal alternating penalty algorithms to solve a wide range class of constrained convex optimization problems. Our approach mainly relies on a novel combination of the classical quadratic penalty, alternating minimization, Nesterov’s acceleration, adaptive strategy for parameters. The first algorithm is designed to solve generic and possibly nonsmooth constrained convex problems without requiring any Lipschitz gradient continuity or strong convexity, while achieving the best-known ( O ( 1 k ) )-convergence rate in a non-ergodic sense, where k is the iteration counter. The second algorithm is also designed to solve non-strongly convex, but semi-strongly convex problems. This algorithm can achieve the best-known ( O ( 1 k^2 ) )-convergence rate on the primal constrained problem. Such a rate is obtained in two cases: (1) averaging only on the iterate sequence of the strongly convex term, or (2) using two proximal operators of this term without averaging. In both algorithms, we allow one to linearize the second subproblem to use the proximal operator of the corresponding objective term. Then, we customize our methods to solve different convex problems, and lead to new variants. As a byproduct, these algorithms preserve the same convergence guarantees as in our main algorithms. We verify our theoretical development via different numerical examples and compare our methods with some existing state-of-the-art algorithms. Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration complexity focus on the special case of proximal gradient method, or accelerated variants thereof. There have been only a few studies of methods that use a second-order approximation to the smooth part, due in part to the difficulty of obtaining closed-form solutions to the subproblems at each iteration. In fact, iterative algorithms may need to be used to find inexact solutions to these subproblems. In this work, we present global analysis of the iteration complexity of inexact successive quadratic approximation methods, showing that an inexact solution of the subproblem that is within a fixed multiplicative precision of optimality suffices to guarantee the same order of convergence rate as the exact version, with complexity related in an intuitive way to the measure of inexactness. Our result allows flexible choices of the second-order term, including Newton and quasi-Newton choices, and does not necessarily require increasing precision of the subproblem solution on later iterations. For problems exhibiting a property related to strong convexity, the algorithms converge at global linear rates. For general convex problems, the convergence rate is linear in early stages, while the overall rate is O(1 k). For nonconvex problems, a first-order optimality criterion converges to zero at a rate of (O(1 k ) ). The accelerated proximal gradient (APG) method, first proposed by Nesterov for minimizing smooth convex functions, later extended by Beck and Teboulle to composite convex objective functions, and studied in a unifying manner by Tseng, has proven to be highly efficient in solving some classes of large scale structured convex optimization (possibly nonsmooth) problems, including nuclear norm minimization problems in matrix completion and @math minimization problems in compressed sensing. The method has superior worst-case iteration complexity over the classical projected gradient method and usually has good practical performance on problems with appropriate structures. In this paper, we extend the APG method to the inexact setting, where the subproblem in each iteration is solved only approximately, and show that it enjoys the same worst-case iteration complexity as the exact counterpart if the subproblems are progressively solved to sufficient accuracy. We apply our inexact APG method to solve large scale ...