Minimizing integrals in certain classes of monotone functions

We solve two continuous extremal problems on the classes of monotone functions: in the first problem we find extremal values for a line integral of a coordinate-wise monotone function of two variables from a rearrangement-invariant class of functions; in the second one we find extremal values for the expectation of a random process with monotone trajectories at a random time. In both cases we reduce the continuous problems to their discrete counterparts. The obtained discrete problems are on the one hand interesting on their own, and on the other hand give a natural explanation of the structure of the extremal functions for the continuous 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.

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 examines properties and interrelations of several concepts of generalized concavity. It shows that the class of functions that are both “generalized concave” and “generalized convex” is closely related to the class of monotone functions of a single variable. After excluding a certain small class of exceptions, the paper shows that, for arbitrary (perhaps not differentiable) functions, concave implies pseudoconcave, pseudoconcave implies strictly quasiconcave, and strictly quasiconcave implies quasiconcave. Several results characterizing the extreme values of generalized concave functions are given.

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.

Partial functions of the k-valued logic monotone with respect to an arbitrary partially ordered set with the least and largest elements and distinct from a lattice are considered. It is shown that the set of closed classes of partial monotone functions containing a precomplete in Pk class of everywhere determined monotone function is infinite.

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.

We consider a convexification method for a class of nonsmooth monotone functions. Specifically, we prove that a semismooth monotone function can be converted into a convex function via certain convexification transformations. The results derived in this paper lay a theoretical base to extend the reach of convexification methods in monotone optimization to nonsmooth situations.

Coudert made a breakthrough in the two-level logic minimization problem with Ordered Binary Decision Diagrams (OBDDs for short) recently [3]. This paper discusses the relationship between the two OBDDs of a monotone function and its prime implicant set to clarify the complexity of this practically efficient method. We show that there exists a monotone function which has an O(n) size sum-of-products but cannot be represented by a polynomial size OBDD. In other words, we cannot obtain the OBDD of the prime implicant set of a monotone function in an output-size sensitive manner once we have constructed the OBDD of that function as in [3], in the worst case. A positive result is also given for a meaningful class of matroid functions.

This paper considers the problem of asymptotic stability in the large of an autonomous system containing a single nonlinearity. The nonlinear function is assumed to belong to several subclasses of monotonically increasing functions in the sector (0, K) , and the stability criterion is shown to be of the form Re Z(j\omega) [G(j\omega) +\frac{1}{K}] -\frac{\delta'}{K} \geq 0 where the constant \delta' is equal to Z(\infty) - Z_{p}(\infty) and Z_{p}(s) is a Popov multiplier. The multiplier Z(s) can, in general, have complex conjugate poles and zeros and is thus more general than the type of multipliers obtained in previous results. The nonlinear functions considered are odd monotonic functions, functions with a power law restriction, and a new class of functions with restricted asymmetry having the property |\frac{f{\theta}{f{\-theta}|\leq c for all \theta . Unlike in certain earlier publications, no upper bound is placed on the derivative of the nonlinearity here. The results obtained can be used to establish stability in some cases even when the Nyquist plot of the linear part transfer function lies in all four quadrants and the nonlinearity is not necessarily odd. Furthermore, it is shown that the conditions on the multiplier and, consequently, those on the linear part can be relaxed as the feedback function approaches linearity.

We show how conditionally negative definite functions on spheres coupled with strictly completely monotone functions (or functions whose derivative is strictly completely monotone) can be used for Hermite interpolation. The classes of functions thus obtained have the advantage over the strictly positive definite functions studied in [17] that closed form representations (as opposed to series expansions) are readily available. Furthermore, our functions include the historically significant spherical multiquadrics. Numerical results are also presented.

We study the recent investigations on a class of functions which are logarithmically completely monotonic. Two open problems are also presented.