On discretization of some extremal problems

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

Minimizing integrals in certain classes of monotone functions

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a Boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010a). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

In this paper, we investigate the lower bound on the number of gates in a Boolean circuit that computes the clique function with a limited number of negation gates. To derive strong lower bounds on the size of such a circuit we develop a new approach by combining three approaches: the restriction applied to constant depth circuits due to Hastad, the approximation method applied to monotone circuits due to Razborov, and the boundary covering developed in the present paper. We prove that if a circuit $C$ with at most $\floor{(1/6) \log \log m}$ negation gates detects cliques of size $(\log m)^{3(\log m)^{1/2}}$ in a graph with $m$ vertices, then $C$ contains at least $2^{(1/5)(\log m)^{(\log m)^{1/2}}}$ gates. No nontrivial lower bounds on the size of such circuits were previously known, even if we restrict the number of negation gates to be a constant. Moreover, it follows from a result of Fischer [{\it Lect. Notes Comput. Sci.,} 33 (1974), pp. 71--82] that if one can improve the number of negation gates from $\floor{(1/6)\log\log m}$ to $\floor{2\log m}$ in the statement, then we have P $\neq$ NP. We also show that the problem of lower bounding the negation-limited circuit complexity can be reduced to the one of lower bounding the maximum of the monotone circuit complexity of the functions in a certain class of monotone functions.

A fixed point theorem is established to prove the existence of a global solution to some fractional differential equations involving the Caputo derivatives under a, not necessarily small, growth condition on the nonlinearity.

This paper deals with the numerical continuation problem of analytic functions $g(z)$ given by $g(z) = \int_\Omega K(z,t) f(t)dt$ where K defines a bounded operator on $L^2 (\Omega )$. It is assumed that g is known over a finite segment A of the real line where g is to be sampled. Our continuation techniques emerged partly from a generalization of the Landweber iteration. We show that a certain discrete approximation of the proposed iterative technique yields two-step (noniterative) algorithms for solving the continuation problem. We also prove convergence of these approximations to the sought function g. Special emphasis on the continuation problem for the case $K(z,t) = e^{ - 2\pi izt} $ is given, and some related numerical examples are presented. The continuation problem when the known part of $g(z)$ is contaminated with some noise is addressed, and some techniques for solving this problem are also provided.

A zero-one linear program is a global discrete optimization problem. Namely, it is a linear programming problem with zero-one variables. The real relaxation of a zero-one program is again a continuous linear programming problem. Simply, the zero-one variables are replaced by continuous variables varying between zero and one independently of each other. If the real relaxation, as a continuous problem, solved with the simplex method happens to have a zero-one optimal solution, then this particular solution is also an optimal solution of the original zero-one programming problem. Suppose that x1,…,xn are all the variables of a given zero-one linear programming problem. For convenience we introduce further variables y1,…,yn defined by y1=1−x1,…, yn=1−xn respectively. For a unified notation the variables x1,…,xn and y1,…, yn together will be denoted by z1,…,z2n. These variables will play the roles the nodes of the conflict graph Г associated with the given zero-one linear programming problem. Fixing the variables zi and zj both to be equal to one simultaneously reduces the number of variables in the problem. If the new smaller optimization problem does not have any feasible solution, then the nodes zi and zj will be connected by an edge in the graph Γ. The graph Γ simply records that there is a conflict between the assignments zi =1 and zj =1 which explains the name conflict graph. In other words the non-directed edge between the nodes zi and zj codes the fact that the inequality zi+zj≤1holds. The conflict graph was designed to generate additional constraints the so-called valid inequalities or cuts in order to expedite the solution of the zero-one linear program via its real relaxation. This particular solution strategy is aptly named the branch and cut method. It will be shown that the conflict graph besides its intended use can also be applied in another solution technique the method of implicit enumeration. We will illustrate by examples that the conflict graph provides us with rules to prune the search tree that are not offered by the commonly applied pruning rules. In addition the computations involved can easily be organized into a highly parallel computational scheme.

Cross-modal hashing is a method which projects heterogeneous multimedia data into a common low-dimensional latent space. Many methods based on hash codes try to keep the relationship between text and corresponding image, and relax the original discrete learning problem into a continuous learning problem. However, these methods may produce ineffective hash codes since they do not make full use of the relationship between different modalities and simply relax the discrete binary constraint into a continuous problem. Collective matrix factorization (CMF) has achieved impressive results in mining semantic concepts or latent topics from image/text. In this paper, we propose a new supervised learning framework which unifies CMF method that maximizes the correlation between two modalities and discrete cyclic coordinate descent (DCC) method that solves NP-hard problems, which ensures that the hash codes generated in the cross-modal are more accurate and efficient. Experiments on three benchmark data sets show the effectiveness of the proposed method.

We are concerned with the numerical solution of distributed optimal control problems for second order elliptic variational inequalities by adaptive finite element methods. Both the continuous problem as well as its finite element approximations represent subclasses of Mathematical Programs with Equilibrium Constraints (MPECs) for which the optimality conditions are stated by means of stationarity concepts in function space (Hintermüller and Kopacka, SIAM J. Optim. 20:868–902, 2009) and in a discrete, finite dimensional setting (Scheel and Scholtes, Math. Oper. Res. 25:1–22, 2000) such as (\(\varepsilon\)-almost, almost) C- and S-stationarity. With regard to adaptive mesh refinement, in contrast to the work in (Hintermüller, ESAIM Control Optim. Calc. Var., 2012, submitted) which adopts a goal oriented dual weighted approach, we consider standard residual-type a posteriori error estimators. The first main result states that for a sequence of discrete C-stationary points there exists a subsequence converging to an almost C-stationary point, provided the associated sequence of nested finite element spaces is limit dense in its continuous counterpart. As the second main result, we prove the reliability and efficiency of the residual-type a posteriori error estimators. Particular emphasis is put on the approximation of the reliability and efficiency related consistency errors by heuristically motivated computable quantities and on the approximation of the continuous active, strongly active, and inactive sets by their discrete counterparts. A detailed documentation of numerical results for two representative test examples illustrates the performance of the adaptive approach.KeywordsA posteriori error analysisElliptic variational inequalitiesFinite elementsOptimal controlStationarity

The study on fractional integrals of confluent hypergeometric functions provides interesting subordination and superordination results and inspired the idea of using this operator to introduce a new class of analytic functions. Given the class of functions An=f∈HU:fz=z+an+1zn+1+…,z∈U written simply A when n=1, the newly introduced class involves functions f∈A considered in the study due to their special properties. The aim of this paper is to present the outcomes of the study performed on the new class, which include a coefficient inequality, a distortion theorem and extreme points of the class. The starlikeness and convexity properties of this class are also discussed, and partial sums of functions from the class are evaluated in order to obtain class boundary properties.

This article explores the algorithms for solving individual classes of discrete extremal problems to find the exact optimum. When obtaining algorithms, the procedures for decoding and finding the maximum upper zero of discrete monotonic functions are used. Formulations of the tasks of decoding and searching for the maximum upper zero of discrete monotonic functions are formulated. To solve the problems of decoding and searching for the maximum upper zero of individual classes of 3-digit monotonic functions, estimates are given for the complexity of deciphering monotonic functions in a class.

The class K of normalized close-to-convex functions in D = {z: z < 1} has a parametric representation involving two Stieltjes integrals. Using a variational method due to G. M. Goluzin (2) for classes of analytic functions defined by a Stieltjes integral, variational formulas are developed for K. With these variational formulas, two general extremal problems within K are solved. The first problem is to maximize the functional Re F(logf'(z)) over K where F(w) is a given entire function and z a given point in D. A special case of this is the rotation theorem for K. The second problem solved is a general coefficient problem. Both problems are solved by characterizing the measures which appear in the integral representation for the extremal functions. The classes of convex univalent functions in D and functions whose derivative has a positive real part in D are proper subclasses of K. The methods used to solve the extremal problems in K can be used for these subclasses as well. Some of the results for the subclasses are known and are not presented here, even though the methods differ from those used previously. It should be mentioned that Goluzin originally used these methods to solve extremal problems of the first type mentioned above within the classes of starlike and typically real functions.

The class A of normalized close-to-convex functions in D={z : \z\ < 1} has a parametric representation involving two Stieltjes integrals.Using a variational method due to G. M. Goluzin [2] for classes of analytic functions defined by a Stieltjes integral, variational formulas are developed for A. With these variational formulas, two general extremal problems within K are solved.The first problem is to maximize the functional Re F[logf'(z)] over A where F(w) is a given entire function and z a given point in D. A special case of this is the rotation theorem for K.The second problem solved is a general coefficient problem.Both problems are solved by characterizing the measures which appear in the integral representation for the extremal functions.The classes of convex univalent functions in D and functions whose derivative has a positive real part in D are proper subclasses of K.The methods used to solve the extremal problems in A can be used for these subclasses as well.Some of the results for the subclasses are known and are not presented here, even though the methods differ from those used previously.It should be mentioned that Goluzin originally used these methods to solve extremal problems of the first type mentioned above within the classes of starlike and typically real functions.

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.