Mean value properties of nondifferentiation functions and their application in nonsmooth analysis

A calculus for (radial) upper convex approximations is developed and mean value properties of nondifferentiable functions are presented. The mean value theorems are formulated with radial upper convex approximations and are of equality type. Some applications in nonsmooth analysis are studied.     

Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations

We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems.     

Decomposition method of descent for minimizing the sum of convex nonsmooth functions

This paper presents a descent method for minimizing a sum of possibly nonsmooth convex functions. Search directions are found by solving subproblems obtained by replacing all but one of the component functions with their polyhedral approximations and adding a quadratic term. The algorithm is globally convergent and terminates when the objective function happens to be polyhedral. It yields a new decomposition method for solving large-scale linear programs with dual block-angular structure.Supported by Program CPBP 02.15.The author thanks the two referees for their helpful suggestions.     

An algorithm for the construction of best approximations based on Kolmogorov's criterion

A descent algorithm for approximating continuous functions having values in a unitary space by functions in a convex class is given. Applications include complex, multidimensional, monotone approximations and the approximation of kernels of integral equations. Numerous complex Chebyshev polynomials are computed numerically.     

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.     

Methods of Chebyshev points of convex sets and their applications

Chebyshev points of bounded convex sets, search algorithms for them, and various applications to convex programming are considered for simple approximations of reachable sets, optimal control, global optimization of additive functions on convex polyhedra, and integer programming. The problem of searching for Chebyshev points in multicriteria models of development and operation of electric power systems is considered.     

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.     

Trilinear Monomials with Mixed Sign Domains: Facets of the Convex and Concave Envelopes

Convex underestimators of nonconvex functions, frequently used in deterministic global optimization algorithms, strongly influence their rate of convergence and computational efficiency. A good convex underestimator is as tight as possible and introduces a minimal number of new variables and constraints. Multilinear monomials over a coordinate aligned hyper-rectangular domain are known to have polyhedral convex envelopes which may be represented by a finite number of facet inducing inequalities. This paper describes explicit expressions defining the facets of the convex and concave envelopes of trilinear monomials over a box domain with bounds of opposite signs for at least one variable. It is shown that the previously used approximations based on the recursive use of the bilinear construction rarely yield the convex envelope itself.     

Convex rearrangements of stable processes

The asymptotic behavior of convex rearrangements for smooth approximations of random processes is considered. The main results are.

- the relations between the convergence of convex rearrangements of absolutely continuous on [0, 1] functions and the weak convergence of its derivatives considered as random variables on the probability space {[0, 1], ß_{[0, 1]}, λ} are established:

- a strong law of large numbers for convex rearrangements of polygonal approximations of stable processes with the exponent α, 1<α≦2, is proved:

- the relations with the results by M. Wshebor (see references) on oscillations of the Wiener process and with the results by Yu. Davydov and A. M. Vershik (see references) on convex rearrangements of random walks are discussed.

     

Line search termination criteria for collinear scaling algorithms for minimizing a class of convex functions

Summary. Many successful quasi-Newton methods for optimization are based on positive definite local quadratic approximations to the objective function that interpolate the values of the gradient at the current and new iterates. Line search termination criteria used in such quasi-Newton methods usually possess two important properties. First, they guarantee the existence of such a local quadratic approximation. Second, under suitable conditions, they allow one to prove that the limit of the component of the gradient in the normalized search direction is zero. This is usually an intermediate result in proving convergence. Collinear scaling algorithms proposed initially by Davidon in 1980 are natural extensions of quasi-Newton methods in the sense that they are based on normal conic local approximations that extend positive definite local quadratic approximations, and that they interpolate values of both the gradient and the function at the current and new iterates. Line search termination criteria that guarantee the existence of such a normal conic local approximation, which also allow one to prove that the component of the gradient in the normalized search direction tends to zero, are not known. In this paper, we propose such line search termination criteria for an important special case where the function being minimized belongs to a certain class of convex functions. Received February 1, 1997 / Revised version received September 8, 1997     

The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed. We study the effect of the use of transformations on the approximation of univariate (convex) functions. In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions. Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations. We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem.     

Quasidifferentiability of nonsmooth quasiconvex functions

The paper deals with nonsmooth quasiconvex functions and develops a quasidifferential analysis for this class of functions. Therefore, in terms of sub and superdifferentials, first order approximations of the functions are derived, optimality conditions are stated and directions of descent (either simple feasible or of steepest descent) are determined. Moreover, a relation among positively homogeneous convex and quasiconvex functions is established     

Model for estimating power shortage in electric power systems with quadratic losses of power in transmission lines

A model for estimating power shortage in electric power systems with quadratic losses of power in transmission lines is studied. This model is designed to analyze problems of reliability of electric power systems. A method of presentation of this model in the form of a convex programming problem is given. An interior point algorithm is proposed for model implementation. This algorithm takes into account quadratic approximations of constraints functions. Results of the experimental study of the algorithm are presented.     

A constraint linearization method for nondifferentiable convex minimization

Summary This paper presents a readily implementable algorithm for solving constrained minimization problems involving (possibly nonsmooth) convex functions. The constraints are handled as in the successive quadratic approximations methods for smooth problems. An exact penalty function is employed for stepsize selection. A scheme for automatic limitation of penalty growth is given. Global convergence of the algorithm is established, as well as finite termination for piecewise linear problems. Numerical experience is reported.Sponsored by Program CPBP 02.15     

Abstract convex approximations of nonsmooth functions

In the article we use abstract convexity theory in order to unify and generalize many different concepts of nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then, we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.     

Limit vector variational inequality problems via scalarization

This paper presents a global optimization approach for solving signomial geometric programming (SGP) problems. We employ an accelerated extended cutting plane (ECP) approach integrated with piecewise linear (PWL) approximations to solve the global optimization of SGP problems. In this approach, we separate the feasible regions determined by the constraints into convex and nonconvex ones in the logarithmic domain. In the nonconvex feasible regions, the corresponding constraint functions are converted into mixed integer linear constraints using PWL approximations, while the other constraints with convex feasible regions are handled by the ECP method. We also use pre-processed initial cuts and batched cuts to accelerate the proposed algorithm. Numerical results show that the proposed approach can solve the global optimization of SGP problems efficiently and effectively.     

Epi-convergence Properties of Smoothing by Infimal Convolution

This paper concerns smoothing by infimal convolution for two large classes of functions: convex, proper and lower semicontinous as well as for (the nonconvex class of) convex-composite functions. The smooth approximations are constructed so that they epi-converge (to the underlying nonsmooth function) and fulfill a desirable property with respect to graph convergence of the gradient mappings to the subdifferential of the original function under reasonable assumptions. The close connection between epi-convergence of the smoothing functions and coercivity properties of the smoothing kernel is established.     

Higher-order total variation bounds for expectations of periodic functions and simple integer recourse approximations

We derive bounds on the expectation of a class of periodic functions using the total variations of higher-order derivatives of the underlying probability density function. These bounds are a strict improvement over those of Romeijnders et al. (Math Program 157:3–46, 2016b), and we use them to derive error bounds for convex approximations of simple integer recourse models. In fact, we obtain a hierarchy of error bounds that become tighter if the total variations of additional higher-order derivatives are taken into account. Moreover, each error bound decreases if these total variations become smaller. The improved bounds may be used to derive tighter error bounds for convex approximations of more general recourse models involving integer decision variables.