About problems of decoding and searching for the maximum upper zero of discrete 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 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.

A computational method for a class of non-standard time optimal control problems involving multiple time horizons

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.

In this paper, we consider the optimal control problem of discrete‐time switched systems. This problem is formulated as an optimization problem involving both continuous and discrete‐valued variables. It can be transformed into a discrete optimization problem. A metric in the space of switching sequences is introduced and an appropriate discrete filled function is constructed. Then, an algorithm that combines the discrete filled function method and a descent method is developed for solving this problem. For illustration, some numerical examples are solved. Copyright © 2009 John Wiley & Sons, Ltd.

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.

A class of coverage growth functions and its practical application

Symplectic difference systems: variable stepsize discretization and discrete quadratic functionals

A new hybrid metaheuristic method for optimizing the objective function on a parallelepiped set of admissible solutions is proposed. It mimics the behavior of a school of river perch when looking for food. The algorithm uses the ideas of several methods: a frog-leaping method, migration algorithms, a cuckoo algorithm and a path-relinking procedure. As an application, a wide class of problems of finding the optimal control of deterministic discrete dynamical systems with a nonseparable performance criterion is chosen. For this class of optimization problems, it is difficult to apply the discrete maximum principle and its generalizations as a necessary optimality condition and the Bellman equation as a sufficient optimality condition. The desire to extend the class of problems to be solved to control problems of trajectory bundles and stochastic problems leads to the need to use not only classical adaptive random search procedures, but also new approaches combining the ideas of migration algorithms and swarm intelligence methods. The efficiency of this method is demonstrated and an analysis is performed by solving several optimal deterministic discrete control problems: two nonseparable problems (Luus–Tassone and LiHaimes) and five classic linear systems control problems with known exact solutions.

Sharp discrete inequalities and applications to discrete variational problems

Discrete facility location problems are classified as types of facility location problems, wherein decisions on choosing facilities in specific locations are made to serve the demand points of customers, thus minimizing the total cost. The covering- and median-based problems are the common classified types of discrete facility location problems, which both comprise different classes of discrete problems as reviewed in this research. However, the discrete facility location problems shown in deterministic and known information and data under uncertain, vague, and ambiguous environments have usually been solved using intuitionistic fuzzy approaches. Neutrosophic is recently applied to tackle the uncertainty and ambiguity of information and data. This paper considered solving the discrete facility location problems under the neutrosophic environment, wherein the information of the locations, distances, times, and costs is uncertain. The mathematical models for the main types of neutrosophic discrete facility location problems, which remain unclear till now despite previous related works, are formulated in this study. Numerical examples demonstrated testing of the neutrosophic discrete models and comparison with the optimization solutions obtained from the normal situations.

Let $T_t$ denote the $t$-threshold function on the $n$-cube: $T_t(x) = 1$ if $|\{i : x_i=1\}| \geq t$, and $0$ otherwise. Define the distance between Boolean functions $g$ and $h$, $d(g,h)$, to be the number of points on which $g$ and $h$ disagree. We consider the following extremal problem: Over a monotone Boolean function $g$ on the $n$-cube with $s$ zeros, what is the maximum of $d(g,T_t)$? We show that the following monotone function $p_s$ maximizes the distance: For $x \in \{0,1\}^n$, $p_s(x)=0$ if and only if $N(x) < s$, where $N(x)$ is the integer whose $n$-bit binary representation is $x$. Our result generalizes the previous work for the case $t=\lceil n/2 \rceil$ and $s=2^{n-1}$ by Blum, Burch, and Langford [BBL98-FOCS98], who considered the problem to analyze the behavior of a learning algorithm for monotone Boolean functions, and the previous work for the same $t$ and $s$ by Amano and Maruoka [AM02-ALT02].

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.