On the strong continuity of convex functions

Heuristic and evolutionary algorithms are proposed to solve challenging real-world optimization problems. In the evolutionary community, many benchmark problems for empirical evaluations of algorithms have been proposed. One of the most important classes of test problems is the class of convex functions, particularly the d-dimensional sphere function. However, the convex function type is somewhat limited. In principle, one can select a set of convex basis functions and use operations that preserve convexity to generate a family of convex functions. This method will inevitably introduce bias in favor of the basis functions. In this paper, the problem is solved by employing insights from computational geometry, which gives the first-ever general-purpose multi-dimensional convex landscape generator. The new proposed generator has the advantage of being generic, which means that it has no bias toward a specific analytical function. A set of N random d-dimensional points is generated for the construction of a d-dimensional convex hull. The upper part of the convex hull is removed by considering the normal of the polygons. The remaining part defines a convex function. It is shown that the complexity of constructing the function is O(Md3), where M is the number of polygons of the convex function. For the method to work as a benchmark function, queries of an arbitrary (d−1) dimensional input are generated, and the generator has to return the value of the convex function. The complexity of answering the query is O(Md). The convexity of the function from the generator is verified with a nonconvex ratio test. The performance of the generator is also evaluated using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) gradient descent algorithm. The source code of the generator is available.

Chapter 9 - Convex and concave functions

In this paper, strongly (α ,T ) -convex functions, i.e., functions f : D → R satisfying the functional inequality f (tx+(1− t)y) t f (x)+(1− t) f (y)− tα(1− t)(x− y)− (1− t)αt(y− x) for x,y ∈ D and t ∈ T ∩ [0,1] are investigated. Here D is a convex set in a linear space, α is a nonnegative function on D−D , and T ⊆ R is a nonempty set. The main results provide various characterizations of strong (α ,T ) -convexity in the case when T is a subfield of R . Mathematics subject classification (2010): Primary 39B62, 26B25.

In this paper, we present a forward---backward splitting algorithm with additional inertial term for solving a strongly convex optimization problem of a certain type. The strongly convex objective function is assumed to be a sum of a non-smooth convex and a smooth convex function. This additional knowledge is used for deriving a worst-case convergence rate for the proposed algorithm. It is proved to be an optimal algorithm with linear rate of convergence. For certain problems this linear rate of convergence is better than the provably optimal worst-case rate of convergence for smooth strongly convex functions. We demonstrate the efficiency of the proposed algorithm in numerical experiments and examples from image processing.

In this paper, we introduce the concepts of m-convex set, mcconvex function and -convex function. We study basic properties for m-convex sets and characterization for such functions.

We start from a basic version of the Hahn-Banach theorem, of which we provide a proof based on Tychonoff’s theorem on the product of compact intervals. Then, in the first section, we establish conditions ensuring the existence of affine functions lying between a convex function and a concave one in the setting of vector spaces — this directly leads to the theorems of Hahn-Banach, Mazur-Orlicz and Fenchel. In the second section, we caracterize those topological vector spaces for which certain convex functions are continuous — this is connected to the uniform boundedness theorem of Banach-Steinhaus and to the closed graph and open mapping theorems of Banach. Combining both types of results readily yields topological versions of the theorems of the first section.KeywordsConvex FunctionTopological Vector SpaceBasic TheoremClosed GraphApply TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this work, we generalized and unified two recent completely different works of Jascha [10] and Lee [2] respectively into one by proposing the proximal stochastic Newton-type gradient (PROXTONE) method for optimizing the sums of two convex functions: one is the average of a huge number of smooth convex functions, and the other is a nonsmooth convex function. Our PROXTONE incorporates second order information to obtain stronger convergence results, that it achieves a linear convergence rate not only in the value of the objective function, but also for the solution. The proofs are simple and intuitive, and the results and technique can be served as a initiate for the research on the proximal stochastic methods that employ second order information. The methods and principles proposed in this paper can be used to do logistic regression, training of deep neural network and so on. Our numerical experiments shows that the PROXTONE achieves better computation performance than existing methods.

In this paper, we present some refinements and reverses for some inequalities involving the weighted arithmetic mean and the weighted geometric mean of two convex functionals. Inequalities involving the Heinz functional mean are also obtained. As applications, we give some refinements and reverses for the relative entropy and the Tsallis relative entropy involving operator or functional arguments.

In this chapter we discuss some properties of convex functions connected with their boundedness and continuity. We start with the following Lemma 6.1.1. Let D ⊂ ℝ N be a convex and open set, and let f : D→ ℝ be a convex function. Then $$ \frac{{f(x) - f(x - nd)}} {n} \leqslant \frac{{f(x) - f(x - md)}} {m} \leqslant \frac{{f(x + md) - f(x)}} {m} \leqslant \frac{{f(x + nd) - f(x)}} {n} $$ for every x ∈ D, d ∈ ℝ N and m, n ∈ ℕ such that 0 < m < n and x ± nd ∈ D.

Nearly strict convexity in Musielak–Orlicz–Bochner function spaces

The work of Banach [1],2 Mazur [2], and others has emphasized the fundamental role played by convex functions and sets in the study of Banach spaces and (continuous) linear transformations between them. It is natural to suspect that similar considerations underlie the theory of general linear transformations, in which closure rather than the restricted notion of continuity is predominant. In this paper, therefore, we study linear transformations T from a Banach space B1 , to a Banach space B2, where T denotes a single-valued, additive and homogeneous operation with linear domain DT in B1, and linear range RT in B2. Our definition of linearity is purely algebraic and will necessarily imply continuity and total definition (DT = B1) cnly in the case of linear functionals. With these exceptions our terminology conforms to that of Banach. When T is linear the expression K(x) = 11 Tx 11, xe DT is a convex function, and E[K(x) _ a], a > 0, is a convex set, where E[K(x) ? a] denotes the set of points x e DK = DT such that K(x) ? a. An examination of the connections between the closure of T, the closure of E[K(x) < a], and the structure of convex K's yields a number of theorems with interesting applications, including as special cases theorems of Gelfand [3] and Bosanquet and Kestelmann [4] of the Banach-Steinhaus type (cf. Zygmund [5], pp. 97-100). Upon specializing B1 and B2 to Hilbert space we obtain a characterization of the domain of a selfadjoint transformation. A by-product of the analysis is a surprisingly simple proof of Wintner's theorem [6, ?108]: every definite symmetric transformation possesses a self-adjoint extension.

In this work we introduce the supra iR-open set in supra topological spaces. , a novel class of supra i-open sets. This class of sets exactly lies between supra regular open and supra i- open sets. We also look at its basic characteristics and compare it to other sorts of data supra semi open sets types as supra i- open set, supra regular open, regular open, regular closed, clopen, supra semi regular open set and supra clopen . By using this set , We introduce and define the notion of supra totally iR-continuous , supra iR-irresolute map and investigate some of its properties. We prove that every supra totally iR- continuous function is supra iR-irresolute, every totally i????-continuous function is supra totally iR continuous and any strongly continuous function is supra totally iR-continuous, on the other hand, we give examples to show that the convers may not be true. At last , We have proved some theorem to discuss the properties of supra iR-normal space and supra iR-regular space.

The paper aims to introduce gβ^-closed sets in bitopological spaces. The relationships of (1, 2)-gβ^-closed sets with other kinds of sets are given. The concept union and intersection of (1,2)- gβ^-closed sets are discussed.Also we introduce (1,2)-gβ^-open sets in this paper. The union and intersection on (1,2)-gβ^-open sets are discussed. Also we give some characterizations of (1, 2)-gβ^-closed sets and (1, 2)-gβ^-open sets in bitopological spaces.

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.

It is shown that, in a non-necessarily Hausdorff real topological vector space, if a subset is a countable disjoint union of convex sets closed in the subset, then those convex sets must be its convex components. On the other hand, by means of convex components we extend the notion of extreme point to non-convex sets, which entails a new equivalent reformulation of the Krein--Milman property (involving drops among other objects). Finally, we study the nature of convex functions and provide some results on their support in order to introduce the concept of multi-slice, that is, slices determined by convex functions (instead of by linear functions). Among other things, we prove that the boundary of a closed convex set with non-empty interior can be obtained as the set of support points of a certain lower semi-continuous convex function on that convex set.