Inexact subgradient methods with applications in stochastic programming

In many instances, the exact evaluation of an objective function and its subgradients can be computationally demanding. By way of example, we cite problems that arise within the context of stochastic optimization, where the objective function is typically defined via multi-dimensional integration. In this paper, we address the solution of such optimization problems by exploring the use of successive approximation schemes within subgradient optimization methods. We refer to this new class of methods as inexact subgradient algorithms. With relatively mild conditions imposed on the approximations, we show that the inexact subgradient algorithms inherit properties associated with their traditional (i.e., exact) counterparts. Within the context of stochastic optimization, the conditions that we impose allow a relaxation of requirements traditionally imposed on steplengths in stochastic quasi-gradient methods. Additionally, we study methods in which steplengths may be defined adaptively, in a manner that reflects the improvement in the objective function approximations as the iterations proceed. We illustrate the applicability of our approach by proposing an inexact subgradient optimization method for the solution of stochastic linear programs.This work was supported by Grant Nos. NSF-DDM-89-10046 and NSF-DDM-9114352 from the National Science Foundation.     

Two “well-known” properties of subgradient optimization

The subgradient method is both a heavily employed and widely studied algorithm for non-differentiable optimization. Nevertheless, there are some basic properties of subgradient optimization that, while “well known” to specialists, seem to be rather poorly known in the larger optimization community. This note concerns two such properties, both applicable to subgradient optimization using the divergent series steplength rule. The first involves convergence of the iterative process, and the second deals with the construction of primal estimates when subgradient optimization is applied to maximize the Lagrangian dual of a linear program. The two topics are related in that convergence of the iterates is required to prove correctness of the primal construction scheme. Dedicated to B.T. Polyak on the occassion of his 70th birthday.     

Subgradient Methods for Saddle-Point Problems

We study subgradient methods for computing the saddle points of a convex-concave function. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. We first present a subgradient algorithm for generating approximate saddle points and provide per-iteration convergence rate estimates on the constructed solutions. We then focus on Lagrangian duality, where we consider a convex primal optimization problem and its Lagrangian dual problem, and generate approximate primal-dual optimal solutions as approximate saddle points of the Lagrangian function. We present a variation of our subgradient method under the Slater constraint qualification and provide stronger estimates on the convergence rate of the generated primal sequences. In particular, we provide bounds on the amount of feasibility violation and on the primal objective function values at the approximate solutions. Our algorithm is particularly well-suited for problems where the subgradient of the dual function cannot be evaluated easily (equivalently, the minimum of the Lagrangian function at a dual solution cannot be computed efficiently), thus impeding the use of dual subgradient methods.     

Subgradient Projectors: Extensions,Theory, and Characterizations

Subgradient projectors play an important role in optimization and for solving convex feasibility problems. For every locally Lipschitz function, we can define a subgradient projector via generalized subgradients even if the function is not convex. The paper consists of three parts. In the first part, we study basic properties of subgradient projectors and give characterizations when a subgradient projector is a cutter, a local cutter, or a quasi-nonexpansive mapping. We present global and local convergence analyses of subgradent projectors. Many examples are provided to illustrate the theory. In the second part, we investigate the relationship between the subgradient projector of a prox-regular function and the subgradient projector of its Moreau envelope. We also characterize when a mapping is the subgradient projector of a convex function. In the third part, we focus on linearity properties of subgradient projectors. We show that, under appropriate conditions, a linear operator is a subgradient projector of a convex function if and only if it is a convex combination of the identity operator and a projection operator onto a subspace. In general, neither a convex combination nor a composition of subgradient projectors of convex functions is a subgradient projector of a convex function.     

Some Results in Nonsmooth Optimization

Some main results in nondifferentiable optimization are reviewed. In section 2, we discuss subgradient methods. Section 3 is about the cutting plane method and the bundle methods are studied in section4. Trust region methods for composite nonsmooth optimization are discussed in section 5.     

Conditional subgradient optimization — Theory and applications

We generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients. Firstly, we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization. Secondly, we consider a particular choice of conditional subgradients, obtained by projections, which leads to an easily implementable modification of traditional subgradient optimization schemes. To evaluate the subgradient projection method we consider its use in three applications: uncapacitated facility location, two-person zero-sum matrix games, and multicommodity network flows. Computational experiments show that the subgradient projection method performs better than traditional subgradient optimization; in some cases the difference is considerable. These results suggest that our simply modification may improve subgradient optimization schemes significantly. This finding is important as such schemes are very popular, especially in the context of Lagrangean relaxation.     

The Extragradient Method for Solving Variational Inequalities in the Presence of Computational Errors

In a Hilbert space, we study the convergence of the subgradient method to a solution of a variational inequality, under the presence of computational errors. Most results known in the literature establish convergence of optimization algorithms, when computational errors are summable. In the present paper, the convergence of the subgradient method for solving variational inequalities is established for nonsummable computational errors. We show that the subgradient method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.     

Abstract convergence theorem for quasi-convex optimization problems with applications

Abstract

Quasi-convex optimization is fundamental to the modelling of many practical problems in various fields such as economics, finance and industrial organization. Subgradient methods are practical iterative algorithms for solving large-scale quasi-convex optimization problems. In the present paper, focusing on quasi-convex optimization, we develop an abstract convergence theorem for a class of sequences, which satisfy a general basic inequality, under some suitable assumptions on parameters. The convergence properties in both function values and distances of iterates from the optimal solution set are discussed. The abstract convergence theorem covers relevant results of many types of subgradient methods studied in the literature, for either convex or quasi-convex optimization. Furthermore, we propose a new subgradient method, in which a perturbation of the successive direction is employed at each iteration. As an application of the abstract convergence theorem, we obtain the convergence results of the proposed subgradient method under the assumption of the Hölder condition of order p and by using the constant, diminishing or dynamic stepsize rules, respectively. A preliminary numerical study shows that the proposed method outperforms the standard, stochastic and primal-dual subgradient methods in solving the Cobb–Douglas production efficiency problem.     

Surrogate Gradient Algorithm for Lagrangian Relaxation

The subgradient method is used frequently to optimize dual functions in Lagrangian relaxation for separable integer programming problems. In the method, all subproblems must be solved optimally to obtain a subgradient direction. In this paper, the surrogate subgradient method is developed, where a proper direction can be obtained without solving optimally all the subproblems. In fact, only an approximate optimization of one subproblem is needed to get a proper surrogate subgradient direction, and the directions are smooth for problems of large size. The convergence of the algorithm is proved. Compared with methods that take effort to find better directions, this method can obtain good directions with much less effort and provides a new approach that is especially powerful for problems of very large size.     

Models and algorithms for the 2-dimensional cell suppression problem in statistical disclosure control

Lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large, structured optimization problems. The dual subgradient scheme does not automatically produce primal feasible solutions; there is an abundance of techniques for computing such solutions (via penalty functions, tangential approximation schemes, or the solution of auxiliary primal programs), all of which require a fair amount of computational effort. We consider a subgradient optimization scheme applied to a Lagrangean dual formulation of a convex program, and construct, at minor cost, an ergodic sequence of subproblem solutions which converges to the primal solution set. Numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained, compared to those generated in the basic subgradient scheme. Received February 11, 1997 / Revised version received June 19, 1998?Published online June 28, 1999     

Nonlinear programming methods in the presence of noise

The problem of minimizing a nonlinear function with nonlinear constraints when the values of the objective, the constraints and their gradients have errors, is studied. This noise may be due to the stochastic nature of the problem or to numerical error.Various previously proposed methods are reviewed. Generally, the minimization algorithms involve methods of subgradient optimization, with the constraints introduced through penalty, Lagrange, or extended Lagrange functions. Probabilistic convergence theorems are obtained. Finally, an algorithm to solve the general convex (nondifferentiable) programming problem with noise is proposed.Originally written for presentation at the 1976 Budapest Symposium on Mathematical Programming.     

Stochastic mirror descent method for distributed multi-agent optimization

This paper considers a distributed optimization problem encountered in a time-varying multi-agent network, where each agent has local access to its convex objective function, and cooperatively minimizes a sum of convex objective functions of the agents over the network. Based on the mirror descent method, we develop a distributed algorithm by utilizing the subgradient information with stochastic errors. We firstly analyze the effects of stochastic errors on the convergence of the algorithm and then provide an explicit bound on the convergence rate as a function of the error bound and number of iterations. Our results show that the algorithm asymptotically converges to the optimal value of the problem within an error level, when there are stochastic errors in the subgradient evaluations. The proposed algorithm can be viewed as a generalization of the distributed subgradient projection methods since it utilizes more general Bregman divergence instead of the Euclidean squared distance. Finally, some simulation results on a regularized hinge regression problem are presented to illustrate the effectiveness of the algorithm.     

A subgradient optimization model for continuous road network design problem

This paper addresses a subgradient optimization model for a well-known continuous network design problem (CNDP). A continuous network design problem can be formulated as a mathematical program with equilibrium constraints (MPEC) where the user equilibrium flows are considered. By contrast to previous studies, in this paper, a conjugate subgradient projection method is presented to efficiently solve the continuous network design problem with global convergence. Numerical calculations are conducted on a real data of road network and various grid-size networks where encouraging results are reported when compared to earlier studies.     

Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization

Inspired by the successful applications of the stochastic optimization with second order stochastic dominance (SSD) model in portfolio optimization, we study new numerical methods for a general SSD model where the underlying functions are not necessarily linear. Specifically, we penalize the SSD constraints to the objective under Slater’s constraint qualification and then apply the well known stochastic approximation (SA) method and the level function method to solve the penalized problem. Both methods are iterative: the former requires to calculate an approximate subgradient of the objective function of the penalized problem at each iterate while the latter requires to calculate a subgradient. Under some moderate conditions, we show that w.p.1 the sequence of approximated solutions generated by the SA method converges to an optimal solution of the true problem. As for the level function method, the convergence is deterministic and in some cases we are able to estimate the number of iterations for a given precision. Both methods are applied to portfolio optimization problem where the return functions are not necessarily linear and some numerical test results are reported.     

Compositions of convex functions and fully linear models

Derivative-free optimization (DFO) is the mathematical study of the optimization algorithms that do not use derivatives. One branch of DFO focuses on model-based DFO methods, where an approximation of the objective function is used to guide the optimization algorithm. Proving convergence of such methods often applies an assumption that the approximations form fully linear models—an assumption that requires the true objective function to be smooth. However, some recent methods have loosened this assumption and instead worked with functions that are compositions of smooth functions with simple convex functions (the max-function or the \(\ell _1\) norm). In this paper, we examine the error bounds resulting from the composition of a convex lower semi-continuous function with a smooth vector-valued function when it is possible to provide fully linear models for each component of the vector-valued function. We derive error bounds for the resulting function values and subgradient vectors.     

A Subgradient Method Based on Gradient Sampling for Solving Convex Optimization Problems

Based on the gradient sampling technique, we present a subgradient algorithm to solve the nondifferentiable convex optimization problem with an extended real-valued objective function. A feature of our algorithm is the approximation of subgradient at a point via random sampling of (relative) gradients at nearby points, and then taking convex combinations of these (relative) gradients. We prove that our algorithm converges to an optimal solution with probability 1. Numerical results demonstrate that our algorithm performs favorably compared with existing subgradient algorithms on applications considered.     

The return plant location problem: Modelling and resolution

The formulation and analysis of a new plant location problem is presented. The problem studied, herein referred to as the Return Plant Location Problem (RPLP), is that of cost minimization in a system of suppliers and customers in which there exists a return product from each customer. Lagrangian decomposition based heuristic and exact solution methods are given. The methods are applied to test problems with different structures and compared with the classical subgradient optimization approach.     

Inexact subgradient methods for quasi-convex optimization problems

In this paper, we consider a generic inexact subgradient algorithm to solve a nondifferentiable quasi-convex constrained optimization problem. The inexactness stems from computation errors and noise, which come from practical considerations and applications. Assuming that the computational errors and noise are deterministic and bounded, we study the effect of the inexactness on the subgradient method when the constraint set is compact or the objective function has a set of generalized weak sharp minima. In both cases, using the constant and diminishing stepsize rules, we describe convergence results in both objective values and iterates, and finite convergence to approximate optimality. We also investigate efficiency estimates of iterates and apply the inexact subgradient algorithm to solve the Cobb–Douglas production efficiency problem. The numerical results verify our theoretical analysis and show the high efficiency of our proposed algorithm, especially for the large-scale problems.     

The effect of deterministic noise in subgradient methods

In this paper, we study the influence of noise on subgradient methods for convex constrained optimization. The noise may be due to various sources, and is manifested in inexact computation of the subgradients and function values. Assuming that the noise is deterministic and bounded, we discuss the convergence properties for two cases: the case where the constraint set is compact, and the case where this set need not be compact but the objective function has a sharp set of minima (for example the function is polyhedral). In both cases, using several different stepsize rules, we prove convergence to the optimal value within some tolerance that is given explicitly in terms of the errors. In the first case, the tolerance is nonzero, but in the second case, the optimal value can be obtained exactly, provided the size of the error in the subgradient computation is below some threshold. We then extend these results to objective functions that are the sum of a large number of convex functions, in which case an incremental subgradient method can be used.     

Variable target value subgradient method

Polyak's subgradient algorithm for nondifferentiable optimization problems requires prior knowledge of the optimal value of the objective function to find an optimal solution. In this paper we extend the convergence properties of the Polyak's subgradient algorithm with a fixed target value to a more general case with variable target values. Then a target value updating scheme is provided which finds an optimal solution without prior knowledge of the optimal objective value. The convergence proof of the scheme is provided and computational results of the scheme are reported.Most of this research was performed when the first author was visiting Decision and Information Systems Department, College of Business, Arizona State University.