In this paper, we examine a class of stochastic optimization problems characterized by nondifferentiability of the objective function. Nondifferentiable optimization deals with problems where the smoothness assumption on the functions is relaxed, meaning that gradients do not necessarily exist. Stochastic optimization problems with nondifferentiable. This problem and techniques to solve it play a central role in contemporary studies in mathematical programming. In this paper, we extend isddp to nondifferentiable msps. Nondifferentiable optimization is a category of optimization that deals with objective that for a variety of reasons is non differentiable and thus nonconvex. Books of clarke and demyanov and vasiliev are devoted to nondi erentiable optimization and book of. Nondifferentiable optimization and polynomial problems nonconvex optimization and its applications pdf,, download. Optimality conditions in fractional semiinfinite multiobjective optimization. This result leads to efficient numerical algorithms that solve the nonconvex problem in the dual domain. The results are subsequently applied to the solution.
The generalization of the steepest descent method for the numerical solution of optimization problems with nondifferentiable cost functions wasgivenbyluenberger 15. Bertsekas nondifferentiable optimization via approximation reader that the class of nondifferentiable problems that we are considering is indeed quite broad. Nondifferentiable multiplier rules for optimization and bilevel optimization problems article pdf available in siam journal on optimization 151. Books of clarke and demyanov and vasiliev are devoted to nondifferentiable optimization and book of korneichuk is devoted to optimization problems of the approximation theory. Convergence of simultaneous perturbation stochastic approximation for nondifferentiable optimization ying he, michael c. The chapter discusses the necessary concepts and the basic properties and some examples of practical problems motivating the use of nso. The basic idea of our approach for numerical solution of problems of the form 1 is to approximate every simple kink in. By contrast the nonlinear programming book focuses primarily on analytical and computational methods for possibly nonconvex differentiable problems.
Chapter vii nondifferentiable optimization sciencedirect. Methods of nondifferentiable and stochastic optimization and. Research article on the application of iterative methods. Use of differentiable and nondifferentiable optimization. Portfolio optimization by minimizing conditional valueatrisk further developed in 25, possesses more appealing features such as subadditivity and convexity, and moreover, it is a coherent risk measure in the sense of artzner et al. Nondifferentiable optimization or nonsmooth optimization nso deals with the situations in operations research where a function that fails to have derivatives for some values of the variables has to be optimized. A descent numerical method for optimization problems with nondifferentiable cost functionals vol 11, no 4 of siam journal of control, 1973. Optimality conditions for nonlinear bilevel vector optimization problems and a global solver can be found in 501 4.
To learn about our use of cookies and how you can manage your cookie settings, please see our cookie policy. Nondifferentiable optimization problems arise in a variety of contexts such as applications in rectilinear data fitting, problems involving euclidean or chebychev norms, and algorithms such as exact penalty methods that change constrained problems into unconstrained problems. In, an inexact variant of stochastic dual dynamic programming sddp called isddp was introduced which uses approximate instead of exact with sddp primal dual solutions of. Convergence of simultaneous perturbation stochastic. On the application of iterative methods of nondifferentiable optimization to some problems of approximation theory stefanm. Minimization methods for nondifferentiable functions 1985. Mitter, a descent numerical method for optirniza tion problems with nondifferentiable cost functionals, siam journal on. Numerical methods for solving nondifferentiable optimization problems, numerical experiments, comparisons and software. Pdf nondifferentiable optimization problems for elliptic. All journal articles featured in optimization vol 25 issue 1. Understand the problem and underline what is important what is known, what is unknown, what we are looking for, dots 2. Interactive bundlebased method for nondifferentiable. Nondifferentiable optimization via approximation mit. This section is devoted to presenting necessarysufficient optimality conditions for fractional semiinfinite multiobjective optimization problems.
Further, we show that the timesharing condition is satisfied for practical multiuser spectrum optimization problems in multicarrier systems in the limit as the number of carriers goes to infinity. Find two positive numbers whose sum is 300 and whose product is a maximum. Nondifferentiable optimization and polynomial problems n. Of recent coinage, the term nondifferentiable optimization ndo covers a spectrum of problems related to finding extremal values of nondifferentiable functions. The term nondifferentiable optimization ndo was introduced by balinski and wolfe 1 for extremum problems with an objective function and constraints that are. Descent methods for composite nondifferentiable optimization. Numerical methods for best chebyshev approximation are suggested, for example, in the book of remez 23. Minimization methods for nondifferentiable functions 1985 by n z shor add to metacart. Marcus abstract in this note, we consider simultaneous perturbation stochastic approximation for function minimization. Each polynomial in n variables can be written as sum of monomials with nonzero coefficients. For example, from the conventional viewpoint, there is no principal difference between functions with continuous gradients which change rapidly and functions with discontinuous gradients. This justifies developing a specialized theory and methods that are the object of this short introduction. Among nonsmooth minimization problems, minimax problems and convex problems have been studied extensively 31, 36, 57, 110, 120. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant fucntions, but for the main results, we give direct proofs based on the properties of the logarithmic function.
Pdf nondifferentiable multiplier rules for optimization. The standard assumption for convergence is that the function be three times. Methods of nondifferentiable and stochastic optimization. For continuous distributions, cvar, also known as the mean excess loss, mean. Ndo problems arise in a variety of contexts, and methods designed for smooth optimization may fail to solve them. The functions in this class of optimization are generally nonsmooth. Stochastic optimization problems with nondifferentiable cost. Nondifferentiable fractional semiinfinite multiobjective. Stochastic optimization problems with nondifferentiable cost functionals 1. The algorithm is based on the classification of objective functions. A nondifferentiable multiobjective optimization problem with nonempty set constraints is considered, and the equivalence of weakly efficient solutions, the critical points for the nondifferentiable multiobjective optimization problems, and solutions for vector variationallike inequalities is established under some suitable conditions.
For nondifferentiable optimization by angelia nedi. A twostage decision problem is shown to give rise to nondifferentiable problems with specific types of nondifferentiability for which simple subgradienttype algorithms are proposed. On nondifferentiable and nonconvex vector optimization. In the sequel, we will often refer to convex ndo, a subclass of nondifferentiable optimization. Portfolio optimization by minimizing conditional valueat. Nondifferentiable optimization of lagrangian dual formulations for linear programs with recovery of primal solutions churlzu lim abstract this dissertation is concerned with solving largescale, illstructured linear programming lp problems via lagrangian dual ld reformulations.
We then combine these cuts with sddp to describe isddp for nondifferentiable msps and analyze the convergence of the method. As noted in the introduction to optimization, an important step in the optimization process is classifying your optimization model, since algorithms for solving optimization problems are tailored to a particular type of problem. Exponential penalty function methods have been used widely in optimization theory by several authors for solving optimization problems of various types see, for example, 2129, and others. The two convex optimization books deal primarily with convex, possibly nondifferentiable, problems and rely on convex analysis. An exponential penalty function method was proposed by murphy 20 for solving nonlinear differentiable scalar optimization problems. In nondifferentiable optimization, the functions may have kinks or corner points, so they cannot be approximated locally by a tangent hyperplane or by a quadratic approximation.
Algorithms for nondifferentiable optimization ladislav luks. We present a random perturbation of the projected variable metric method for solving linearly constrained nonsmooth i. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The investigation of bilevel optimization problems with fuzzy lower level problems can be found in 390, 596, 757, 22, 18, 18. Apr 16, 2020 all journal articles featured in optimization vol 25 issue 1. Nurminski the problem of optimal control for the nonlinear dynamic system with discrete time is considered. Abstract, in this paper, we examine a class of stochastic optimiza tion problems characterized by nondifferentiability of the objective function. The approach is to replace the original problem by an approximate one which. Find two positive numbers whose product is 750 and for which the sum of one and 10 times the other is a minimum. It is shown that, in many cases, the expected value of the objective function is differentiable and, thus, the resulting optimization problem can be solved by using classical analytical or numerical methods. Abstract pdf 473 kb 2014 nonsmooth algorithms and nesterovs smoothing technique for generalized fermattorricelli problems. Bertsekas nondifferentiable optimization via approximation nonlinear constraints or they are applicable only to a special class of problems such as minimax problems of particular form. Random perturbation of the projected variable metric. Progress in nondifferentiable optimization core reader.
This paper makes progress toward solving optimization problems of this type by showing that under a certain condition called the timesharing condition, the duality gap of the optimization problem is always zero, regardless of the. Pdf on nondifferentiable and nonconvex vector optimization. This paper presents three general schemes for extending differentiable optimization algorithms to nondifferentiable problems. Siam journal on optimization society for industrial and. Papers of andersen, calamai and conn, overton,andxueandye consider minimization of sum of euclidean norms. On the application of iterative methods of nondifferentiable. It is shown that, in many cases, the expected value of. We introduce a smoothing technique for nondifferentiable optimization problems. A local randomized smoothing technique farzad youse.
Pdf we introduce a new method for solving a class of nonsmooth unconstrained optimization problems. The paper tries to develop the basic features of the two main direct approaches in ndo, namely the subgradient concept. Nondifferentiable optimization ndo also called nonsmooth optimization nso concerns problems in which the functions involved have discontinuous first derivatives. On the mathematical foundations of nondifferentiable optimization.
Nimbus, an interactive method for nondifferentiable multiobjective optimization problems, is described. We first provide formulas for inexact cuts for value functions of convex nondifferentiable optimization problems. The basic idea of our approach for numerical solution of problems of the form 1 is to approximate every simple kink in the functional expression. Varayia abstract, in this paper, we examine a class of stochastic optimiza tion problems characterized by nondifferentiability of the objective function. Semiinfinite optimization algorithms, nondifferentiable optimization. Here we provide some guidance to help you classify your optimization model. Optimization online inexact cuts in sddp applied to.
This volume contains selected papers presented at the workshop. Nondifferentiable, also known as nonsmooth, optimization ndo is concerned with problems where the smoothness assumption on the functions involved is relaxed. Contact problems of two elastic or elastoplastic plates with. This chapter discusses the nondifferentiable optimization ndo. Inexact cuts in sddp applied to multistage stochastic nondifferentiable problems. Bertsekas 2 abstract we consider a class of subgradient methods for minimizing a convex function that consists of the sum of a large number of component functions. This type of minimization arises in a dual context from lagrangian relaxation of the coupling constraints of. Using a nondifferentiable penalty function it is possible to transform the initial problem into an unconditional one. We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. Stochastic optimization problems with nondifferentiable cost functionals 1 d, p. Nondifferentiable optimization or nonsmooth optimization nso deals with the situations in operations research where a function that fails to have derivatives for some values of the variables has. Nondifferentiability means that the gradient does not exist, implying that the function may have kinks or corner points.
Nondifferentiable optimization via approximation vol 1, no 25 of mathematical programming study 3, 1975. On nondifferentiable and nonconvex vector optimization problems article pdf available in journal of optimization theory and applications 1063. Shanbhag abstractwe consider a class of stochastic nondifferentiable optimization problems where the objective function is an expectation of a random convex function, that is not necessarily differentiable. Nondifferentiable optimization and polynomial problems. Ek academy of sciences of the czech republic we present four basic fortran subroutines for nondifferentiable optimization with simple bounds and general linear constraints.
Nondifferentiable optimization and polynomial problems nonconvex optimization and its applications pdf,, download ebookee alternative. The gconvergence approach for nondifferentiable optimization problems was used by a. Pdf a method for nondifferentiable optimization problems. Numerical methods for best chebyshev approximation are suggested, for example, in the book. Some convergence results are given and the method is illustrated by means of examples from nonlinear programming. Pdf we present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. Bertsekas, stochastic optimization problems with nondifferentiable cost functionals, journal of optimization theory and applications 12 pp. It is shown that the armijo gradient method, phaseiphaseii methods of feasible directions and exact penalty function methods have conceptual analogs for problems with locally lipschitz functions and implementable. It is shown that the armijo gradient method, phaseiphaseii methods of feasible directions and exact penalty function methods have conceptual analogs for problems with locally lipschitz functions and implementable analogs for problems with semismooth functions. The links between nondifferentiable optimization and structured decisionmaking problems are considered in the paper by a.
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