Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

A Lyusternik–Graves theorem for the proximal point method

We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion y∈T(x), where y is a perturbation element near 0 and T is a set-valued mapping acting from a Banach space X to a Banach space Y which is metrically regular around some point $({\bar{x}},0)$ in its graph. We study the behavior of the convergent iterates generated by the algorithm and we prove that they inherit the regularity properties of T, and vice versa. We analyze the cases when the mapping T is metrically regular and strongly regular.     

Local properties of functions and approximation by trigonometric polynomials

Suppose Φ_{p, E} (p>0 an integer, E ?[0, 2π]) is a family of positive nondecreasing functions? _x(t) (t>0, x∈ E) such that? _x(nt)≤n^P ? _x(t) (n=0,1,...), t_n is a trigonometric polynomial of order at most n, and Δ _h ^l (f, x) (l>0 an integer) is the finite difference of orderl with step h of the functionf.THEOREM. Supposef (x) is a function which is measurable, finite almost everywhere on [0, 2π], and integrable in some neighborhood of each point xε E,? _X εΦ_p,E and $$\overline {\mathop {\lim }\limits_{\delta \to \infty } } |(2\delta )^{ - 1} \smallint _{ - \delta }^\delta \Delta _u^l (f,x)du|\varphi _x^{ - 1} (\delta ) \leqslant C(x)< \infty (x \in E).$$ . Then there exists a sequence {t _n } _n=1 ^∞ which converges tof (x) almost everywhere, such that for x ε E $$\overline {\mathop {\lim }\limits_{n \to \infty } } |f(x) - l_n (x)|\varphi _x^{ - 1} (l/n) \leqslant AC(x),$$ where A depends on p andl.     

Finite-dimensional quasi-variational inequalities associated with discontinuous functions

In this paper, given a nonempty closed convex setX ? ⁿ , a functionf: X→? ⁿ , and a multifunction Γ:X→2^X, we deal with the problem of finding a point \(\hat x\) ∈X such that $$\hat x \in \Gamma (\hat x) and \langle f(\hat x), \hat x - y\rangle \leqslant 0, for all y \in \Gamma (\hat x).$$ For such problem, we establish a result where, in particular, the functionf is not assumed to be continuous. More precisely, we extend to the present setting a finite-dimensional version of a result by Ricceri on variational inequalities (Ref. 1).     

Estimation in nonparametric location-scale regression models with censored data

Consider the random vector (X, Y), where X is completely observed and Y is subject to random right censoring. It is well known that the completely nonparametric kernel estimator of the conditional distribution ${F(\cdot|x)}$ of Y given X = x suffers from inconsistency problems in the right tail (Beran 1981, Technical Report, University of California, Berkeley), and hence any location function m(x) that involves the right tail of ${F(\cdot|x)}$ (like the conditional mean) cannot be estimated consistently in a completely nonparametric way. In this paper, we propose an alternative estimator of m(x), that, under certain conditions, does not share the above inconsistency problems. The estimator is constructed under the model Y = m(X) + σ(X)ε, where ${\sigma(\cdot)}$ is an unknown scale function and ε (with location zero and scale one) is independent of X. We obtain the asymptotic properties of the proposed estimator of m(x), we compare it with the completely nonparametric estimator via simulations and apply it to a study of quasars in astronomy.     

Sensitivity of Solutions to a Parametric Generalized Equation

In the present paper we prove that, under some suitable conditions on multifunctions $C\!: {I\!\!R}^l\rightrightarrows {I\!\!R}^n$ , $F\!:{I\!\!R}^d\times {I\!\!R}^n\rightrightarrows {I\!\!R}^m$ , and $K\!:{I\!\!R}^l\times {I\!\!R}^n\rightrightarrows {I\!\!R}^m$ , the generalized perturbation multifunction $G\!: {I\!\!R}^d\times{I\!\!R}^l\times {I\!\!R}^m\rightrightarrows {I\!\!R}^n$ , of the form $$G(\mu,\lambda,\nu)=\{x\in C(\lambda)\ |\ \nu\in F(\mu,x)+K(\lambda,x)\},$$ is proto-differentiable at (μ, λ, ν) relative to x?∈?G(μ, λ, ν). Moreover, in a special case, where K(λ, x) is a normal cone to C(λ) at x, we also provide sufficient conditions for G(·) to be single-valued on a neighborhood of (μ, λ, ν) and semi-differentiable at (μ, λ, ν).     

Restricted Weak Upper Semi-continuity of Subdifferentials of Convex Functions on Banach Spaces

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X ^* there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with $g \leqslant f$ on X is generically Fréchet differentiable. If, in addition, $\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty$ , then X is an Asplund space.     

Properties of vertex packing and independence system polyhedra

We consider two convex polyhedra related to the vertex packing problem for a finite, undirected, loopless graphG with no multiple edges. A characterization is given for the extreme points of the polyhedron \(\mathcal{L}_G = \{ x \in R^n :Ax \leqslant 1_m ,x \geqslant 0\} \) , whereA is them × n edge-vertex incidence matrix ofG and 1 _m is anm-vector of ones. A general class of facets of = convex hull{x∈R ⁿ :Ax≤1 _m ,x binary} is described which subsumes a class examined by Padberg [13]. Some of the results for are extended to a more general class of integer polyhedra obtained from independence systems.     

Fast Integration for Cauchy Principal Value Integrals of Oscillatory Kind

In this paper we give a simple, but high order and rapid convergence method for computing the Cauchy principal value integrals of the form $\int_{-1}^{1}e^{i\omega x}\frac{f(x)}{x-\tau}dx$ and its error bounds, where f(x) is a given smooth function, ω∈R ⁺ may be large and ?1<τ<1. The proposed method is constructed by approximating $(\frac{f(x)-f(\tau)}{x-\tau})^{(s)}$ by using the special Hermite interpolation polynomial, which is a Taylor series. The validity of the method has been demonstrated by the results of several numerical experiments and the comparisons with other methods.     

Hereditary order convexity inL(X,Y)

LetX be a topological vector space,Y an ordered topological vector space andL(X,Y) the space of all linear and continuous mappings fromX intoY. The hereditary order-convex cover [K] ^h of a subsetK ofL(X,Y) is defined by [K] ^h ={A∈L(X,Y):Ax∈[Kx] for allx∈X}, where[Kx] is the order-convex ofKx. In this paper we study the hereditary order-convex cover of a subset ofL(X,Y). We show how this cover can be constructed in specific cases and investigate its structural and topological properties. Our results extend to the spaceL(X,Y) some of the known properties of the convex hull of subsets ofX ^*.     

An interior point potential reduction algorithm for the linear complementarity problem

The linear complementarity problem (LCP) can be viewed as the problem of minimizingx ^T y subject toy=Mx+q andx, y?0. We are interested in finding a point withx ^T y <ε for a givenε > 0. The algorithm proceeds by iteratively reducing the potential function $$f(x,y) = \rho \ln x^T y - \Sigma \ln x_j y_j ,$$ where, for example,ρ=2n. The direction of movement in the original space can be viewed as follows. First, apply alinear scaling transformation to make the coordinates of the current point all equal to 1. Take a gradient step in the transformed space using the gradient of the transformed potential function, where the step size is either predetermined by the algorithm or decided by line search to minimize the value of the potential. Finally, map the point back to the original space. A bound on the worst-case performance of the algorithm depends on the parameterλ ^*=λ^*(M, ε), which is defined as the minimum of the smallest eigenvalue of a matrix of the form $$(I + Y^{ - 1} MX)(I + M^T Y^{ - 2} MX)^{ - 1} (I + XM^T Y^{ - 1} )$$ whereX andY vary over the nonnegative diagonal matrices such thate ^T XYe ?ε andX _jj Y _jj?n ². IfM is a P-matrix,λ ^* is positive and the algorithm solves the problem in polynomial time in terms of the input size, |log ε|, and 1/λ ^*. It is also shown that whenM is positive semi-definite, the choice ofρ = 2n+ \(\sqrt {2n} \) yields a polynomial-time algorithm. This covers the convex quadratic minimization problem.     

Cohomological q-convexity versus local q-completeness with corners

We show that if (D, π) is an unramified Riemann domain over a distinguished complex manifold X such that D is cohomologically q-convex, then π is locally q-complete with corners. We call X distinguished if for every point x of X there is a holomorphic line bundle $\cal L$ on X (which may depend on x) so that the global sections $\Gamma (X \cal L)$ of $\cal L$ generate its 1-jets at x. Examples of distinguished complex manifolds include all complex submanifolds of C^m × Pⁿ; in particular all Stein or projectively algebraic manifolds.     

Weak real integral characterizations for exponential stability of semigroups in reflexive spaces

Let (t _n ) be a sequence of nonnegative real numbers tending to ∞, such that 1≤t _n+1?t _n ≤α for all natural numbers n and some positive α. We prove that a strongly continuous semigroup {T(t)} _t≥0, acting on a Hilbert space H, is uniformly exponentially stable if $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, y\bigr\rangle\bigr|\bigr)<\infty, $$ for all unit vectors x, y in H. We obtain the same conclusion under the assumption that the inequality $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, x^\ast\bigr\rangle\bigr|\bigr)<\infty, $$ is fulfilled for all unit vectors x∈X and x ^?∈X ^?, X being a reflexive Banach space. These results are stated for functions φ belonging to a special class of functions, such as defined in the second section of this paper. We conclude our paper with a Rolewicz’s type result in the continuous case on Hilbert spaces.     

CGS algorithms for constrained extremization of functions

CGS (conjugate Gram-Schmidt) algorithms are given for computing extreme points of a real-valued functionf(x) ofn real variables subject tom constraining equationsg(x)=0,M<n. The method of approach is to solve the system $$\begin{gathered} f'(x) + g'(x)*\lambda = 0 \hfill \\ g(x) = 0 \hfill \\ \end{gathered} $$ where λ is the Lagrange multiplier vector, by means of CGS algorithms for systems of nonlinear equations. Results of the algorithms applied to test problems are given, including several problems having inequality constraints treated by adjoining slack variables.     

Multivalued Exponentiation Analysis. Part III: Forward Exponentials

We continue our work on exponentiability of multivalued maps on Banach spaces. In Part I we studied the exponentiability of a map F : X ? X by using a Maclaurin expansion approach. In Part II we studied the recursive exponentiation approach. Recursive exponentials are built by using the trajectories of a discrete-time evolution system governed by F. We now focus the attention on forward exponentiability. The forward exponential of F at the point x is defined as the Kuratowski limit $$e^{F}(x):=\lim\limits_{n\to\infty}\; \left(I+ n^{-1} F \right)^{n}(x).$$ This type of exponential arises in connection with the Euler discretization scheme for solving the first-order differential inclusion \(\dot \psi (t)\in F(\psi (t))\) .     

Spectral singularities of Klein-Gordon s-wave equations with an integral boundary condition

We investigate the spectral singularities and the eigenvalues of the boundary value problem $$\begin{gathered} y'' + \left[ {\lambda - Q\left( x \right)} \right]^2 y = 0,x \in R_ + = [0,\infty ), \hfill \\ \quad \int\limits_0^\infty {K\left( x \right)y\left( x \right)dx + \alpha y'\left( 0 \right) - \beta y\left( 0 \right) = 0,} \hfill \\ \end{gathered}$$ where Q and K are complex valued functions, K∈L ²(R ₊), α,β∈C with |α|+|β|≠0 and λ is a spectral parameter.     

Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.     

Finitely Additive Equivalent Martingale Measures

Let L be a linear space of real bounded random variables on the probability space $(\varOmega ,\mathcal{A},P_{0})$ . There is a finitely additive probability P on $\mathcal{A}$ such that P～P ₀ and E _P (X)=0 for all X∈L if and only if cE _Q (X)≤ess?sup?(?X), X∈L, for some constant c>0 and (countably additive) probability Q on $\mathcal{A}$ such that Q～P ₀. A necessary condition for such a P to exist is $\overline{L-L_{\infty}^{+}}\cap L_{\infty}^{+}=\{0\}$ , where the closure is in the norm-topology. If P ₀ is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability P on $\mathcal{A}$ such that P?P ₀ and E _P (X)=0 for all X∈L if and only if ess?sup?(X)≥0 for all X∈L.