Convexification of Nonsmooth Monotone Functions1

At least since 1904 we know about the existence of singular monotone functions, i.e. continuous monotone functions of one real variable with vanishing derivative a.e. In that year Lebesgue [7] and Minkowski [8] gave their famous examples. In 1910 Faber [4], in 1916 Sierpinski [10] and then many others greatly enriched the variety of known examples. By integrating anyone of these functions we get a differentiable convex function with vanishing second derivative a.e. Thus we obtain smooth convex curves with vanishing curvature a.e. A very beautiful and simple example is due to de R h a m [3J: Take a convex polygon, consider on each side the two points dividing it into three equal parts and take the convex polygon with all these division points as vertices. Repeat the procedure. As a limit we get a smooth, strictly convex curve with a.e. vanishing curvature! The purpose of this paper is to show that, in the sense of Baire categories, most convex curves have the above property. More generally, most convex surfaces in IR" have a.e. a vanishing sectional curvature in every tangent direction. We start with some definitions. By a convex surface we shall always understand a closed convex surface in Busemann's sense (see [11, p. 3). A convex curve is a one-dimensional convex surface. A point x of a convex surface S is called smooth if S is differentiable at x. S is smooth if each of its points is smooth. Any half-line in a supporting plane (this is a hyperplane, see [1], p. 4) of a convex surface S, originating at a point x ~ S will be called supporting direction at x. A supporting direction at a smooth point is called tangent direction. The union of a circular with a square 2-cell, such that the radius and centre of the circle coincide with the side-length and a vertex of the square, is called corner-disk. The two points lying on both the circle and the square are called touching points of the corner-disk. A 2-cell having the union of half a circle with a segment as boundary is called semidisk. The points of the boundary of a semidisk which are not smooth are called corners of the semidisk. The radius of a corner-disk or a semidisk is the radius of the circle appearing in their definitions.

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.

The theory of evolutionary computation (EC) has experienced rapid and productive growth in recent years. New proof techniques and novel theoretical frameworks have allowed advances in our understanding of the processes and structures inherent in evolutionary optimization. As a result, the frontiers of our knowledge have been expanded further than ever before. Some recent trends in this field, which are covered in this issue, include developments in the understanding of the behavior of evolutionary algorithms (EAs) in dynamic environments rather than just static settings, a theoretical appreciation of the advantages arising from the parallelization of evolutionary algorithms through a greater comprehension of the underlying dynamics, and an understanding of algorithm behavior on broad function classes, including -hard problems.The primary goal of this special issue is to provide extended and polished versions of diverse examples of the best theoretical work presented at conferences in 2014, and to serve as a forum for researchers to advance the theoretical understanding of evolutionary computation methods. The papers included in this special issue span a plurality of topics and offer the reader a cross section of recent outstanding work in EC theory.In dynamic optimization the objective function changes over time, and optimization algorithms face the additional challenge of tracking these changes to be successful. The article “Analysis of Randomised Search Heuristics for Dynamic Optimisation,” by Thomas Jansen and Christine Zarges, presents a novel analytical framework for the analysis of randomized search heuristics on dynamic problems inspired by the fixed-budget computations perspective. The authors introduce a new interesting class of bi-stable dynamic functions where the optimum oscillates between two complementary strings, and apply the framework to analyze and compare the performance of evolutionary algorithms and artificial immune systems on the novel class of functions.Over three decades ago, László Lovász observed that in discrete optimization, submodularity is the counterpart to convexity. However, in contrast to the focus on convex functions in continuous evolutionary optimization, so far submodular functions have received comparatively little attention from EC theoreticians studying discrete functions. The article “Maximizing Submodular Functions under Matroid Constraints by Evolutionary Algorithms,” by Tobias Friedrich and Frank Neumann, addresses this gap by analyzing the performance of evolutionary algorithms on different classes of submodular functions. The maximization of submodular functions is -hard in general, and the authors present several approximation results for monotone submodular and nonmonotone symmetric submodular functions under different kinds of matroid constraints.The idea behind parallel evolutionary algorithms is to evolve multiple subpopulations in parallel and allow interprocess communication at given time intervals. During these migration phases, fractions of each subpopulation can be shared among the subpopulations. There is very little understanding of how the migration frequency affects algorithmic performance, so setting the migration interval parameter appropriately may be difficult. In the article “Design and Analysis of Schemes for Adapting Migration Intervals in Parallel Evolutionary Algorithms,” Andrea Mambrini and Dirk Sudholt propose two schemes to automatically adapt the migration interval of parallel EAs during execution and provide a rigorous analytical framework that yields upper bounds on the expected runtime and expected communication effort of the parallel EAs with different migration topologies for various function classes.In the field of genetic programming, a long-standing open problem is how to address the issue of bloat, the emergence during evolution of solution elements that do not contribute significantly or at all to program fitness or semantics but increase program complexity. The article “On the Performance of Different Genetic Programming Approaches for the SORTING Problem,” by Markus Wagner, Frank Neumann, and Tommaso Urli, tackles the issue of bloat control in the context of sorting. As a basis for their study, they consider program trees and use some measure of sortedness of an in-order traversal to evaluate their fitness. The authors investigate single- and multiobjective variants of genetic programming algorithms with and without bloat control mechanisms, give rigorous upper bounds on their running times, and complement the study with experiments.The topic of constraint handling has recently gained traction in the continuous domain. The article “Markov Chain Analysis of Cumulative Step-Size Adaptation on a Linear Constrained Problem,” by Alexandre Chotard, Anne Auger, and Nikolaus Hansen, presents a rigorous analysis of a (1,)-Evolution Strategy using resampling on a linear function with a linear constraint. The authors prove the previously assumed property that a Markov chain, describing the behavior of the algorithm, exhibits stability in cases with constant step-size and with cumulative step-size adaptation with cumulation parameter equal to 1. This property characterizes the divergence of the algorithm with constant step-size and the geometric divergence or convergence with step-size adaptation, implying fast convergence of Monte Carlo simulations of the divergence rate.In their seminal 2006 paper, Droste, Jansen, and Wegener introduced the concept of black box complexity in order to establish a complexity theory for general-purpose randomized search heuristics. Generally speaking, the black box complexity of a problem is a lower bound on the number of function evaluations needed by any black box algorithm to solve it. Recently, black box models have been refined and developed extensively, allowing the hardness of objective function classes to be more precisely understood. The article “Unbiased Black Box Complexities of Jump Functions,” by Benjamin Doerr, Carola Doerr, and Timo Kötzing, analyzes the unbiased black box complexity of a Jump function class where in each function a local optimum is k bits in distance from the global optimum. The authors provide polynomial upper bounds on the black box complexity of Jump for different sizes of the gap. In particular, they show that an unbiased polynomial-time black box algorithm exists even when almost all of the search space is a plateau of constant fitness.The guest editors would like to thank the authors for their contributions, the referees for their careful reviewing and constructive comments, and the editor-in-chief, Hans-Georg Beyer, for his support in preparing this special issue.

There is a strong correlation between convexity and symmetry concepts. In this study, we investigated the new generic class of functions called the (n,m)–generalized convex and studied its basic algebraic properties. The Hermite–Hadamard inequality for the (n,m)–generalized convex function, for the products of two functions and of this type, were proven. Moreover, this class of functions was applied to several known identities; midpoint-type inequalities of Ostrowski and Simpson were derived. Our results are extensions of many previous contributions related to integral inequalities via different convexities.

In this paper, we introduce notable Jensen–Mercer inequality for a general class of convex functions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with some examples. As a result, we establish Hermite–Jensen–Mercer inequalities pertaining uniformly convex functions by considering the class of fractional integral operators. Moreover, we establish Mercer–Ostrowski inequalities for conformable integral operator via differentiable uniformly convex functions. Finally, we apply our inequalities to get estimations for normal probability distributions (Gaussian distributions).

We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank k (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k. Every monotone function that has supermodular degree k (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(k+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-k.One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k+1 if all players hold valuation functions in MPH-k. The other is an upper bound of 2k on the price of anarchy of simultaneous first price auctions.

Let I be a finite interval and r,s∈N. Given a set M, of functions defined on I, denote by Δ+sM the subset of all functions y∈M such that the s-difference Δτsy(⋅) is nonnegative on I, ∀τ>0. Further, denote by Δ+sWpr, the class of functions x on I with the seminorm ‖x(r)‖Lp≤1, such that Δτsx≥0, τ>0. Let Mn(hk):={∑i=1ncihk(wit−θi)∣ci,wi, θi∈R, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions hk(t)=t+k, t∈R, k∈N0. We give two-sided estimates both of the best unconstrained approximation E(Δ+sWpr,Mn(hk))Lq, k=r−1,r, s=0,1,...,r+1, and of the best s-monotonicity preserving approximation E(Δ+sWpr,Δ+sMn(hk))Lq, k=r−1,r, s=0,1,...,r+1. The most significant results are contained in theorem 2.2.

Order preserving and order reversing operators on the class of convex functions in Banach spaces

If f:C → R is a nonnegative, monotone decreasing function from the n-dimensional cube C:= [0, 1]n into the real numbers with \(\int\limits_\text{C} {\text{f} > 0}\), then one can find two convex functions g1 ≤ f ≤ g2 such that for k > O the following inequalities hold. $$\prod\limits_{\text{i} = 2}^{\text{n} + 1} {(\frac{{\text{k + i}}} {\text{i}})^{1/\text{k}} \text{M}_\text{k} \text{(g}_\text{1} ) \geqslant \text{M}_\text{k} \text{(f)} \geqslant \text{(n + 1)}^{\text{ - (k + n + 1)/k}} \text{M}_\text{k} (\text{g}_\text{2} ),}$$ where \(\text{M}_\text{k} (\text{f}): = \left( {\frac{1} {{\text{k + 1}}}\,\frac{{\int\limits_\text{C} {\text{f}^{\text{k} + 1} } }} {{\int\limits_\text{C} \text{f} }}} \right)^{1/\text{k}}\)is called the k-moment of f.

Fejér’s integral inequality is a weighted version of the Hermite-Hadamard inequality that holds for the class of convex functions. To derive his inequality, Fejér [Über die Fourierreihen, II, Math. Naturwiss, Anz. Ungar. Akad. Wiss. 24 (1906), 369–390] assumed that the weight function is symmetric w.r.t. the midpoint of the interval. In this study, without assuming any symmetry condition on the weight function, Fejér’s inequality is extended to the class of sub-biharmonic functions, namely, the set of functions f ∈ C 4 ( I ) f\in {C}^{4}\left({\mathbb{I}}) satisfying f ″ ″ ≤ 0 {f}^{^{\prime\prime} ^{\prime\prime} }\le 0 , where I {\mathbb{I}} is an interval of R {\mathbb{R}} . In the special case when the weight function is symmetric w.r.t. the midpoint of some interval, and the function f f is convex and sub-biharmonic, an interesting refinement of Fejér’s inequality is deduced. Moreover, this inequality is extended to a new class of functions defined on the plane, that we call the set of sub-biharmonic functions on the coordinates. Assuming in addition that the function is convex on the coordinates, a refinement of an inequality due to Dragomir [On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan. J. Math. 5 (2001), 775–788] is obtained.

Given a continuous function f defined on the unit cube ofR n and a convex function φt, φt(0)=0, φt(x)>0, for x>0, we prove that the set of best Lφt-approximations by monotone functions has exactly one element ft, which is also a continuous function. Moreover if the family of convex functions {φt}t>0 converges uniformly on compact sets to a function φ0, then the best approximation ft→f0 uniformly, as t→0, where f0 is the best approximation of f within the Orlicz space L0. The best approximations {ft} are obtained as well as minimizing integrals or the Luxemburg norm.

A convex function f : [a,b] → ℝ satisfies the so-called Hermite-Hadamard inequality f (a + b 2 ) ≤ 1 b − a∫ abf(t)dt ≤f(a) + f(b) 2 . Motivated by the above estimates, in this paper we consider approximately monotone and convex functions, and give upper and lower bounds to the numerical integral mean, i.e., to 1 b−aIn(f), where In(f) denotes some of the most popular Newton-Cotes quadrature formulas.

It is the purpose of this paper to propose a novel class of convex functions associated with strong η-convexity. A relationship between the newly defined function and an earlier generalized class of convex functions is hereby established. To point out the importance of the new class of functions, some examples are presented. Additionally, the famous Hermite–Hadamard inequality is derived for this generalized family of convex functions. Furthermore, some inequalities and results for strong η-convex function are also derived. We anticipate that this new class of convex functions will open the research door to more investigations in this direction.

In this paper we consider a class of structured nonsmooth difference-of-convex (DC) minimization in which the first convex component is the sum of a smooth and nonsmooth functions while the second convex component is the supremum of possibly infinitely many convex smooth functions. We first propose an inexact enhanced DC algorithm for solving this problem in which the second convex component is the supremum of finitely many convex smooth functions, and show that every accumulation point of the generated sequence is an $$(\alpha ,\eta )$$ -D-stationary point of the problem, which is generally stronger than an ordinary D-stationary point. In addition, inspired by the recent work (Pang et al. in Math Oper Res 42(1):95–118, 2017; Wen et al. in Comput Optim Appl 69(2):297–324, 2018), we propose two proximal DC algorithms with extrapolation for solving this problem. We show that every accumulation point of the solution sequence generated by them is an $$(\alpha ,\eta )$$ -D-stationary point of the problem, and establish the convergence of the entire sequence under some suitable assumption. We also introduce a concept of approximate $$(\alpha ,\eta )$$ -D-stationary point and derive iteration complexity of the proposed algorithms for finding an approximate $$(\alpha ,\eta )$$ -D-stationary point. In contrast with the DC algorithm (Pang et al. 2017), our proximal DC algorithms have much simpler subproblems and also incorporate the extrapolation for possible acceleration. Moreover, one of our proximal DC algorithms is potentially applicable to the DC problem in which the second convex component is the supremum of infinitely many convex smooth functions. In addition, our algorithms have stronger convergence results than the proximal DC algorithm in Wen et al. (2018).

In this paper a sharp upper bound of second Hankel determinant 2 2 4 3 a a − a for the functions belonging to the class M (A, B) α of alpha convex functions is established. The class of alpha convex functions is an extended version of the classes of starlike functions and convex functions. By giving the particular values to alpha, it is easy to obtain the results of starlike and convex functions. The class discussed in this paper is an extended version of the class of alpha convex functions. By giving the particular values to A and B, the result of alpha convex functions can be easily obtained.