The minimal criterion for the equivalence between local and global optimal solutions in nondifferentiable optimization problem

In the paper, a necessary and sufficient criterion it provided such that any local optimal solution is also global in a not necessarily differentiable constrained optimization problem. This criterion is compared to others earlier appeared in the literature, which are sufficient but not necessary for a local optimal solution to be global. The importance of the established criterion is illustrated by suitable examples of nonconvex optimization problems presented in the paper.     

Completely reducible super-simple designs with block size four and related super-simple packings

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A super-simple design \({\mathcal{D}}\) with point set X, block set \({\mathcal{B}}\) and index λ is called completely reducible super-simple (CRSS), if its block set \({\mathcal{B}}\) can be written as \({\mathcal{B}=\bigcup_{i=1}^{\lambda} \mathcal{B}_i}\), such that \({\mathcal{B}_i}\) forms the block set of a design with index unity but having the same parameters as \({\mathcal{D}}\) for each 1 ≤ i ≤ λ. It is easy to see, the existence of CRSS designs with index λ implies that of CRSS designs with index i for 1 ≤ i ≤ λ. CRSS designs are closely related to q-ary constant weight codes (CWCs). A (v, 4, q)-CRSS design is just an optimal (v, 6, 4)_q+1 code. On the other hand, CRSS group divisible designs (CRSSGDDs) can be used to construct q-ary group divisible codes (GDCs), which have been proved useful in the constructions of q-ary CWCs. In this paper, we mainly investigate the existence of CRSS designs. Three neat results are obtained as follows. Firstly, we determine completely the spectrum for a (v, 4, 3)-CRSS design. As a consequence, a class of new optimal (v, 6, 4)₄ codes is obtained. Secondly, we give a general construction for (4, 4)-CRSSGDDs with skew Room frames, and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDD of type g ^u are also sufficient except definitely for \({(g,u)\in \{(2,4),(3,4),(6,4)\}}\). Finally, we consider the related optimal super-simple (v, 4, 2)-packings and show that such designs exist for all v ≥ 4 except definitely for \({v\in \{4,5,6,9\}}\).     

Covering arrays for some equivalence classes of words

Covering arrays for words of length over a ‐letter alphabet are arrays with entries from the alphabet so that for each choice of columns, each of the ‐letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case known as partitioning hash families, words are equivalent if they induce the same partition of a element set. In the second case, words of the same weight are equivalent. In both cases, we produce logarithmic upper bounds on the minimum size of a covering array. Definitive results for , as well as general results, are provided.     

The optimal statistical median of a convex set of arrays

We consider the following problem. A set of vectors is given. We want to find the convex combination such that the statistical median of z is maximum. In the application that we have in mind, are the historical return arrays of asset j and are the portfolio weights. Maximizing the median on a convex set of arrays is a continuous non-differentiable, non-concave optimization problem and it can be shown that the problem belongs to the APX-hard difficulty class. As a consequence, we are sure that no polynomial time algorithm can ever solve the model, unless P = NP. We propose an implicit enumeration algorithm, in which bounds on the objective function are calculated using continuous geometric properties of the median. Computational results are reported.     

Optimum detecting arrays for independent interaction faults

The use of detecting arrays (DTAs) is motivated by the need to locate and detect interaction faults arising between the factors in a component-based system in software testing. The optimality and construction of DTAs have been investigated in depth for the case in which all the interaction faults are assumed to have the same strength; however, as a practical concern, the strengths of these faults are not always identical. For real world applications, it would be desirable for a DTA to be able to identify and detect faulty interactions of a strength equal to or less than a specified value under the assumption that the faulty interactions are independent from one another. To the best of our knowledge, the optimality and construction of DTAs for independent interaction faults have not been studied systematically before. In this paper, we establish a general lower bound on the size of DTAs for independent interaction faults and explore the combinatorial feature that enable these DTAs to meet the lower bound. Taking advantage of this combinatorial characterization, several classes of optimum DTAs meeting the lower bound are presented.     

The equivalence between pointwise Hardy inequalities and uniform fatness

We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions.     

On the affine equivalence relation between two classes of Boolean functions with optimal algebraic immunity

Recently, two classes of Boolean functions with optimal algebraic immunity have been proposed by Carlet et al. and Wang et al., respectively. Although it appears that their methods are very different, it is proved in this paper that the two classes of Boolean functions are in fact affine equivalent. Moreover, the number of affine equivalence classes of these functions is also studied.     

The equivalence between processor sharing and service in random order

In this note we explore a useful equivalence relation for the delay distribution in the G/M/1 queue under two different service disciplines: (i) processor sharing (PS); and (ii) random order of service (ROS). We provide a direct probabilistic argument to show that the sojourn time under PS is equal (in distribution) to the waiting time under ROS of a customer arriving to a non-empty system. We thus conclude that the sojourn time distribution for PS is related to the waiting-time distribution for ROS through a simple multiplicative factor, which corresponds to the probability of a non-empty system at an arrival instant. We verify that previously derived expressions for the sojourn time distribution in the M/M/1 PS queue and the waiting-time distribution in the M/M/1 ROS queue are indeed identical, up to a multiplicative constant. The probabilistic nature of the argument enables us to extend the equivalence result to more general models, such as the M/M/1/K queue and ·/M/1 nodes in product-form networks.     

The equivalence of “strong calmness” and “calmness” in optimal control theory

In an earlier paper optimality conditions expressed in terms of a Lipschitz continuous function which satisfies a condition resembling the Hamilton-Jacobi-Bellman equation were derived. It was shown that a certain hypothesis, strong calmness, is the weakest hypothesis under which such developments are possible. In the present paper it is shown that strong calmness is equivalent to calmness for a wide class of problems. Support is thereby given to strong calmness as being a reasonable hypothesis, since calmness is apparently the weakest known hypothesis assuring normality, in the sense that the Pontryagin maximum principle applies with the cost multiplier nonzero.     

The equivalence of optimal perspective formulation and Shor's SDP for quadratic programs with indicator variables

In this paper, we compare the strength of the optimal perspective relaxation and Shor's SDP relaxation for convex quadratic optimization problems with indicator variables and prove that they are equivalent.     

Constructions of optimal orthogonal arrays with repeated rows

We construct orthogonal arrays ${\mathsf{OA}}_{\lambda }(k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m∕\lambda$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k\ge n+1$, albeit with large $\lambda$. We also study basic OAs; these are optimal OAs in which $gcd(m,\lambda )=1$. We construct a basic OA with $n=2$ and $k=4t+1$, provided that a Hadamard matrix of order $8t+4$ exists. This completely solves the problem of constructing basic OAs with $n=2$, modulo the Hadamard matrix conjecture.     

A sign test for detecting the equivalence of Sylvester Hadamard matrices

In this paper we show that every matrix in the class of Sylvester Hadamard matrices of order 2 ^k under H-equivalence can have full row and column sign spectrum, meaning that tabulating the numbers of sign interchanges along any row (or column) gives all integers 0,1,...,2 ^k  − 1 in some order. The construction and properties of Yates Hadamard matrices are presented and is established their equivalence with the Sylvester Hadamard matrices of the same order. Finally, is proved that every normalized Hadamard matrix has full column or row sign spectrum if and only if is H-equivalent to a Sylvester Hadamard matrix. This provides us with an efficient criterion identifying the equivalence of Sylvester Hadamard matrices.     

An equivalence between local fields

The p-component of the index of a number field K depends only on the completions of K at the primes over p. In this paper we define an equivalence relation between m-tuples of local fields such that, if two number fields K and K^′ have equivalent m-tuples of completions at the primes over p, then they have the same p-component of the index. This equivalence can be interpreted in terms of the decomposition groups of the primes over p of the normal closures of K and K^′.     

The equivalence between the T-stabilities of Mann and Ishikawa iterations

We show that T-stability of Mann and Ishikawa iterations are equivalent.     

The equivalence between the TE,TM model and the Darwin models

In this paper,we study the TE,TM model by using the decompositions of the vector fields in 2-D bounded multiply connected domains and 2-D unbounded domains,respectively.We find that the TE,TM model and the Darwin models are equivalent if we assume some regularity of the initial data.     

The equivalence of linear codes implies semi-linear equivalence

Designs, Codes and Cryptography - We prove that if two linear codes are equivalent then they are semi-linearly equivalent. We also prove that if two additive MDS codes over a field are equivalent...     

The equivalence principle

For the example of the motion of an accelerated charge it is shown that an inertial frame of reference in which there is a homogeneous static gravitational field with strength g is physically inequivalent to a uniformly accelerated frame of reference moving with acceleration-g with respect to the inertial frame of reference. It follows from this that the equivalence principle does not hold in the usual formulation. The widely held opinion that such a principle is the basis of the general theory of relativity is not entirely correct. Einstein's theory of gravitation is based on an equivalence principle of a deeper content, which takes the form that the metric fieldg of a Riemannian space is declared to be a gravitational field. Such is the natural formulation of the equivalence principle at which Einstein subsequently arrived.Institute of High Energy Physics; Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 1, pp. 121–140, April, 1994.