Generalized Convex Functions Research Articles

Orlicz Uzayının Tanımı Üzerine Bir Not

The Orlicz spaces were introduced by Z.W. Birnbaum and W. Orlicz in 1931 as a natural generalization of the classical Lebesgue spaces. For this generalization the function �ݔ ௣ � entering in the definition of Lebesgue's space is replaced by a more general convex function Ф. Until today, Orlicz spaces have been studied by many authors from a different aspects. But, most of these studies used different definitions for the convex function Ф to define the Orlicz spaces. In this paper we will investigate the exact relationships between these different definitions of convex functions Ф and obtain the differences between the corresponding Orlicz spaces.

Inner γ-Convex Functions in Normed Linear Spaces

A real-valued function f defined on a convex subset D of some normed linear space X is said to be inner γ-convex w.r.t. some fixed roughness degree γ > 0 if there is a ν ∈ [0, 1] such that $$\sup \limits _{\lambda \in [2, 1+1/\nu ]} \left (f((1-\lambda )x_{0}+\lambda x_{1})- (1-\lambda )f(x_{0})- \lambda f(x_{1})\right )\geq 0 $$ holds for all x 0, x 1 ∈ D satisfying ∥x 0 − x 1∥ = ν γ and −(1/ν)x 0 + (1 + 1/ν)x 1 ∈ D. The requirement of this kind of roughly generalized convex functions is very weak; nevertheless, they also possess properties similar to those of convex functions relative to their supremum. For instance, if an inner γ-convex function defined on some bounded convex subset D of an inner product space attains its maximum, then it has maximizers at some strictly γ-extreme points of D. In this paper, some sufficient conditions and examples for γ-convex functions and several properties relative to the location of their maximizers are given.

On the Convergence of Decomposition Methods for Multistage Stochastic Convex Programs

We prove the almost-sure convergence of a class of sampling-based nested decomposition algorithms for multistage stochastic convex programs in which the stage costs are general convex functions of the decisions and uncertainty is modelled by a scenario tree. As special cases, our results imply the almost-sure convergence of stochastic dual dynamic programming, cutting-plane and partial-sampling (CUPPS) algorithm, and dynamic outer-approximation sampling algorithms when applied to problems with general convex cost functions.

An asynchronous parallel stochastic coordinate descent algorithm

We describe an asynchronous parallel stochastic coordinate descent algorithm for minimizing smooth unconstrained or separably constrained functions. The method achieves a linear convergence rate on functions that satisfy an essential strong convexity property and a sublinear rate (1/K) on general convex functions. Near-linear speedup on a multicore system can be expected if the number of processors is O(n1/2) in unconstrained optimization and O(n1/4) in the separable-constrained case, where n is the number of variables. We describe results from implementation on 40-core processors.

Asynchronous Stochastic Coordinate Descent: Parallelism and Convergence Properties

We describe an asynchronous parallel stochastic proximal coordinate descent algorithm for minimizing a composite objective function, which consists of a smooth convex function added to a separable convex function. In contrast to previous analyses, our model of asynchronous computation accounts for the fact that components of the unknown vector may be written by some cores simultaneously with being read by others. Despite the complications arising from this possibility, the method achieves a linear convergence rate on functions that satisfy an optimal strong convexity property and a sublinear rate ($1/k$) on general convex functions. Near-linear speedup on a multicore system can be expected if the number of processors is $O(n^{1/4})$. We describe results from implementation on 10 cores of a multicore processor.

The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results

After studying finite asymptotic expansions in real powers, we have developed a general theory for expansions of type (*) ,x → x0 where the ordered n-tuple forms an asymptotic scale at x0 , i.e. as x → x0, 1 ≤ i ≤ n – 1, and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x o. As in previous papers by the author concerning polynomial, real-power and two-term theory, the locution “factorizational theory” refers to the special approach based on various types of factorizations of a differential operator associated to . Moreover, the guiding thread of our theory is the property of formal differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and n -1 expansions obtained by formal applications of suitable linear differential operators of orders 1,2,…,n-1. Some considerations lead to restrict the attention to two sets of operators naturally associated to “canonical factorizations”. This gives rise to conjectures whose proofs build an analytic theory of finite asymptotic expansions in the real domain which, though not elementary, parallels the familiar results about Taylor’s formula. One of the results states that to each scale of the type under consideration it remains associated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion(*), if valid, is automatically formally differentiable n-1 times in two special senses.

Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-B: Solutions of Differential Inequalities and Asymptotic Admissibility of Standard Derivatives

Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*) , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x0. The main result states that to each scale of this type it remains as-sociated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n ? 1 times in the two special senses characterized in Part II-A. A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.

Spontaneous Stochasticity and Anomalous Dissipation for Burgers Equation

We develop a Lagrangian approach to conservation-law anomalies in weak solutions of inviscid Burgers equation, motivated by previous work on the Kraichnan model of turbulent scalar advection. We show that the entropy solutions of Burgers possess Markov stochastic processes of (generalized) Lagrangian trajectories backward in time for which the Burgers velocity is a backward martingale. This property is shown to guarantee dissipativity of conservation-law anomalies for general convex functions of the velocity. The backward stochastic Burgers flows with these properties are not unique, however. We construct infinitely many such stochastic flows, both by a geometric construction and by the zero-noise limit of the Constantin–Iyer stochastic representation of viscous Burgers solutions. The latter proof yields the spontaneous stochasticity of Lagrangian trajectories backward in time for Burgers, at unit Prandtl number. It is conjectured that existence of a backward stochastic flow with the velocity as martingale is an admissibility condition which selects the unique entropy solution for Burgers. We also study linear transport of passive densities and scalars by inviscid Burgers flows. We show that shock solutions of Burgers exhibit spontaneous stochasticity backward in time for all finite Prandtl numbers, implying conservation-law anomalies for linear transport. We discuss the relation of our results for Burgers with incompressible Navier–Stokes turbulence, especially Lagrangian admissibility conditions for Euler solutions and the relation between turbulent cascade directions and time-asymmetry of Lagrangian stochasticity.

Extreme point inequalities and geometry of the rank sparsity ball

We investigate geometric features of the unit ball corresponding to the sum of the nuclear norm of a matrix and the $$l_1$$l1 norm of its entries--a common penalty function encouraging joint low rank and high sparsity. As a byproduct of this effort, we develop a calculus (or algebra) of faces for general convex functions, yielding a simple and unified approach for deriving inequalities balancing the various features of the optimization problem at hand, at the extreme points of the solution set.

Existence result for the motion of several rigid bodies in an incompressible non-Newtonian fluid with growth conditions in Orlicz spaces

We prove the existence of weak solutions to the problem of the motion of one or several rigid bodies immersed in a non-Newtonian fluid of rheology more general than power-law. The nonlinear viscous term in the equation is described with the help of a general convex function defining Orlicz spaces. The main ingredient of the proof is the convergence of the nonlinear term achieved with the help of pressure localization and monotonicity methods.

SOLVING A MINIMIZATION PROBLEM FOR A CLASS OF CONSTRAINED MAXIMUM EIGENVALUE FUNCTION

Nonsmooth convex optimization problem is a class of important problems in operational research. Bundle methods are considered as one of the most efficient methods for solving nonsmooth optimization problems. The methods have already been applied to many practical problems. In this paper, using bundle method, the optimization problem that the sum of maximum eigenvalue function and general non-smooth convex function can be solved. Through approximation to the objective function, the proximal bundle method based on approximate model is given. We prove that the sequences generated by the algorithm converge to the optimal solution of the original problem. Finally, the algorithm is used to solve a class of constrained maximum eigenvalue function.

Generalized Convex Functions on Fractal Sets and Two Related Inequalities

We introduce the generalized convex function on fractal setsRα (0<α≤1)of real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen’s inequality and generalized Hermite-Hadamard's inequality. Furthermore, some applications are given.

Variational Inequalities and Vector Optimization 2014

In the last three decades, the theory of variational analysis provides very effective and powerful techniques for studying a wide class of problems arising in nonlinear equations, optimization problems, economics equilibrium, game theory, complementarity problems, and fixed point problems, as well as other branches of mathematics and engineering sciences. So, the thorough study of both theory and methods about variational inequalities will help us to find new techniques for solving the practical problem. Vector optimization problems have received much attention by many authors due to their extensive applications in many fields such as biology, economics, optimal control, and differential inclusions. Because of the importance and active impact of the variational inequality and the vector optimization problem in the nonlinear analysis and optimization, this special issue, focusing on most recent contributions, includes works on variational inequalities, equilibrium problems and nonexpansive mappings, vector optimization problems and generalized convex functions, robust optimization problems, and optimization problems with applications, which are based on a strict international peer review procedure and our original proposal. A brief review of the papers is given under the following four topics.

A Proximal Stochastic Gradient Method with Progressive Variance Reduction

We consider the problem of minimizing the sum of two convex functions: one is the average of a large number of smooth component functions, and the other is a general convex function that admits a simple proximal mapping. We assume the whole objective function is strongly convex. Such problems often arise in machine learning, known as regularized empirical risk minimization. We propose and analyze a new proximal stochastic gradient method, which uses a multistage scheme to progressively reduce the variance of the stochastic gradient. While each iteration of this algorithm has similar cost as the classical stochastic gradient method (or incremental gradient method), we show that the expected objective value converges to the optimum at a geometric rate. The overall complexity of this method is much lower than both the proximal full gradient method and the standard proximal stochastic gradient method.

On Stochastic Subgradient Mirror-Descent Algorithm with Weighted Averaging

This paper considers stochastic subgradient mirror-descent method for solving constrained convex minimization problems. In particular, a stochastic subgradient mirror-descent method with weighted iterate-averaging is investigated and its per-iterate convergence rate is analyzed. The novel part of the approach is in the choice of weights that are used to construct the averages. Through the use of these weighted averages, we show that the known optimal rates can be obtained with simpler algorithms than those currently existing in the literature. Specifically, by suitably choosing the stepsize values, one can obtain the rate of the order $1/k$ for strongly convex functions, and the rate $1/\sqrt{k}$ for general convex functions (not necessarily differentiable). Furthermore, for the latter case, it is shown that a stochastic subgradient mirror-descent with iterate averaging converges (along a subsequence) to an optimal solution, almost surely, even with the stepsize of the form $1/\sqrt{1+k}$, which was not p...

A Parallel Bundle Framework for Asynchronous Subspace Optimization of Nonsmooth Convex Functions

An algorithmic framework is presented for optimizing general convex functions by nonsynchronized parallel processes. Each process greedily picks a suitable adaptive subset of coordinates and runs a bundle method on a corresponding restricted problem stopping whenever a descent step is encountered or predicted decrease is reduced sufficiently. No prior knowledge on the dependencies between variables is assumed. Instead, dependency information is collected automatically by analyzing aggregate subgradient properties required for ensuring convergence. Within this framework three strategies are discussed for supporting varying scenarios of structural independence: a single convex function, the sum of partially separable convex functions, and a scenario tuned to problem decomposition by Lagrangian relaxation of packing-type constraints. The theoretical framework presented here generalizes a practical method proposed by the authors for Lagrangian relaxation of large scale packing problems and simplifies the analysis.

Brownian Inventory Models with Convex Holding Cost, Part 1: Average-Optimal Controls

We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed cost and a proportional cost. The challenge is to find an adjustment policy that balances the holding cost and adjustment cost to minimize the long-run average cost. When both upward and downward fixed costs are positive, our problem is an impulse control problem. When both fixed costs are zero, our problem is a singular or instantaneous control problem. For the impulse control problem, we prove that a four-parameter control band policy is optimal among all feasible policies. For the singular control problem, we prove that a two-parameter control band policy is optimal. We use a lower-bound approach, widely known as the “verification theorem”, to prove the optimality of a control band policy for both the impulse and singular control problems. Our major contribution is to prove the existence of a “smooth” solution to the free boundary problem under some mild assumptions on the holding cost function. The existence proof leads naturally to a numerical algorithm to compute the optimal control band parameters. We demonstrate that the lower-bound approach also works for a Brownian inventory model which prohibits an inventory backlog. A companion paper (“Brownian inventory models with convex holding cost, part 2: discount-optimal controls”, Stochastic Systems, Vol. 3, No. 2, pp. 500–573) explains how to adapt the lower-bound approach to study a Brownian inventory model under a discounted cost criterion.

Brownian Inventory Models with Convex Holding Cost, Part 2: Discount-Optimal Controls

We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed positive cost and a proportional cost. The challenge is to find an adjustment policy that balances the inventory cost and adjustment cost to minimize the expected total discounted cost. We provide a tutorial on using a three-step lower-bound approach to solving the optimal control problem under a discounted cost criterion. In addition, we prove that a four-parameter control band policy is optimal among all feasible policies. A key step is the constructive proof of the existence of a unique solution to the free boundary problem. The proof leads naturally to an algorithm to compute the four parameters of the optimal control band policy.

KOROVKIN SETS FOR SEQUENCES OF SHAPE-PRESERVING OPERATORS

This paper examines Korovkin sets for sequences of operators which are shape-preserving relative to a cone of generalized convex functions, and gives some applications to shape-preserving linear approximation.

SECOND-ORDER UNIVEX FUNCTIONS AND GENERALIZED DUALITY MODELS FOR MULTIOBJECTIVE PROGRAMMING PROBLEMS CONTAINING ARBITRARY NORMS

In this paper, we introduce three new broad classes of second-order generalized convex functions, namely, (<TEX>$\mathcal{F}$</TEX>, <TEX>$b$</TEX>, <TEX>${\phi}$</TEX>, <TEX>${\rho}$</TEX>, <TEX>${\theta}$</TEX>)-sounivex functions, (<TEX>$\mathcal{F}$</TEX>, <TEX>$b$</TEX>, <TEX>${\phi}$</TEX>, <TEX>${\rho}$</TEX>, <TEX>${\theta}$</TEX>)-pseudosounivex functions, and (<TEX>$\mathcal{F}$</TEX>, <TEX>$b$</TEX>, <TEX>${\phi}$</TEX>, <TEX>${\rho}$</TEX>, <TEX>${\theta}$</TEX>)-quasisounivex functions; formulate eight general second-order duality models; and prove appropriate duality theorems under various generalized (<TEX>$\mathcal{F}$</TEX>, <TEX>$b$</TEX>, <TEX>${\phi}$</TEX>, <TEX>${\rho}$</TEX>, <TEX>${\theta}$</TEX>)-sounivexity assumptions for a multiobjective programming problem containing arbitrary norms.