A free coordination hull approach to efficiency measurement

Superficially, the free disposal hull production possibility set (pps) can be regarded as a binary version of the Banker, Charnes and Cooper pps, in which only one component of the vector, λ, is non-zero and of necessity equal to 1. We therefore propose, by analogy with a pps due originally to Koopmans, a new pps which retains the binary characteristic of the components of λ but permits more than one component to be non-zero. Thus, a given DMU's performance is assessed not only against the individual DMUs included in the sample but also against composite DMUs obtained by simple aggregation. This approach is demonstrated and investigated on a published data set.     

Global attractor for a nonlinear thermoviscoelastic model with a non-convex free energy density

This paper is concerned with the existence of a global attractor for a semiflow governed by the weak solutions to a nonlinear one-dimensional thermoviscoelasticity with a non-convex free energy density. The constitutive assumptions for the Helmholtz free energy include the model for the study of martensitic phase transitions in shape memory alloys. To describe physically phase transitions between different configurations of crystal lattices, we work in a framework in which the strain u$u$ belongs to L^∞$L^{\infty }$. New approaches are introduced and more delicate estimates are derived to establish the crucial L^∞$L^{\infty }$-estimate of strain u$u$ in deriving the compactness of the orbit of the semiflow and the existence of an absorbing set.     

The inverse hull of the free semigroup on a set X

In this paper the inverse hull of the free semigroup on a non-empty set X is characterized. It is shown that if F^O is a certain semilattice and (G,*) is the free group on X, then is isomorphic to a Rees factor semigroup of a subsemigroup of (F^O x G, #), where (A,w)#(B,u) = ((A*w B)*w^–1, w*u). Several properties of , including Green's relations on are also determined.     

Global existence for a quasi-linear evolution equation with a non-convex energy

We establish the existence of global in time weak solutions to the initial-boundary value problem related to the dynamics of coherent solid-solid phase transitions in viscoelasticity. The class of the stored energy functionals includes the double well potential, and a general convolution damping term is considered.

     

Stability and optimal control of thermomechanical processes with non-convex free energies of Ginzburg–Landau type

In this paper we study a coupled non-linear system of partial differential equations that models the dynamics of structural phase transitions in a one-dimensional non-viscous and heat-conducting solid. The corresponding Helmholtz free energy density is assumed in Ginzburg–Landau form; to allow for phase transitions and hysteresis phenomena, it is not assumed convex in the order parameter. It is shown that the solution of the system depends continuously upon the data, and we prove an existence result for an associated optimal control problem.     

Functional inequalities on manifolds with non-convex boundary

In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary.     

Differentiability properties for a class of non-convex functions

Closed sets K ⊂ satisfying an external sphere condition with uniform radius (called ϕ-convexity or proximal smoothness) are considered. It is shown that for -a.e. x ∊ ∂K the proximal normal cone to K at x has dimension one. Moreover if K is the closure of an open set satisfying a (sharp) nondegeneracy condition, then the De Giorgi reduced boundary is equivalent to ∂ K and the unit proximal normal equals -a.e. the (De Giorgi) external normal. Then lower semicontinuous functions f : with ϕ-convex epigraph are shown, among other results, to be locally BV and twice -a.e. differentiable; furthermore, the lower dimensional rectifiability of the singular set where f is not differentiable is studied. Finally we show that for -a.e. x there exists δ (x) > 0 such that f is semiconvex on B(x,δ(x)). We remark that such functions are neither convex nor locally Lipschitz, in general. Methods of nonsmooth analysis and of geometric measure theory are used. Work partially supported by M.I.U.R., project “Viscosity, metric, and control theoretic methods for nonlinear partial differential equations.”     

Convex drawings of graphs with non-convex boundary constraints

In this paper, we study a new problem of convex drawing of planar graphs with non-convex boundary constraints, and call a drawing in which every inner-facial cycle is drawn as a convex polygon an inner-convex drawing. It is proved that every triconnected plane graph with the boundary fixed with a star-shaped polygon whose kernel has a positive area admits an inner-convex drawing. We also prove that every four-connected plane graph whose boundary is fixed with a crown-shaped polygon admits an inner-convex drawing. We present linear time algorithms to construct inner-convex drawings for both cases.     

基于一种新的集合插值的非凸紧集的细分概型

对A rtstein给出的度量平均的定义作了改进,给出一种新的集合插值,并基于这种新的集合插值,对相应的关于一般紧集的样条细分和插值细分分别作了研究,并给出了细分的收敛性性质.与此同时,将这种新的集合插值与基于度量平均的插值及基于M inkow sk i平均的插值分别作了比较,可以看出新的集合插值在某些方面具有更好的物理性质.     

Fenchel-Rockafellar type duality for a non-convex non-differential optimization problem

A Fenchel-Rockafellar type duality theorem is obtained for a non-convex and non-differentiable maximization problem by embedding the original problem in a family of perturbed problems. The recent results of Ivan Singer are developed in this more general framework. A relationship is also established between the solutions and optimal values of the primal and dual problems using the theory of subdifferential calculus.     

Efficiency measurement for parallel production systems

In the real world there are systems which are composed of independent production units. The conventional data envelopment analysis (DEA) model uses the sum of the respective inputs and outputs of all component units of a system to calculate its efficiency. This paper develops a parallel DEA model which takes the operation of individual components into account in calculating the efficiency of the system. A property owned by this parallel model is that the inefficiency slack of the system can be decomposed into the inefficiency slacks of its component units. This helps the decision maker identify inefficient components and make subsequent improvements. Another property is that the efficiency calculated from this model is smaller than that calculated from the conventional DEA model. Few systems will have perfect efficiency score; consequently, a stronger discrimination power is gained. In addition to theoretical derivations, a case of the national forests of Taiwan is used as an example to illustrate the whole idea.     

A proximal iterative approach to a non-convex optimization problem

We consider a variable Krasnosel’skii-Mann algorithm for approximating critical points of a prox-regular function or equivalently for finding fixed-points of its proximal mapping prox_λf. The novelty of our approach is that the latter is not non-expansive any longer. We prove that the sequence generated by such algorithm (via the formula x_k+1=(1−α_k)x_k+α_kprox_λkfx_k, where (α_k) is a sequence in (0,1)), is an approximate fixed-point of the proximal mapping and converges provided that the function under consideration satisfies a local metric regularity condition.     

A solution formula for a non-convex scalar hyperbolic conservation law with monotone initial data

In this paper we prove an explicit representation formula for the solution of a one-dimensional hyperbolic conservation law with a non-convex flux function but monotone initial data. This representation formula is similar to those of Lax [10] and Kunik [7,8] and enables us to compute the solution pointwise explicitly. This result is a generalization of a theorem given in Kunik [8] where the case of only one inflexion point for the fluxes was considered. Its proof uses the polygonal method of Dafermos [2]. The application of this method leads to a simple explicit construction of the solutions for a Kynch sedimentation process [9] and to an explicit parameter representation for the shock curves evolving during the sedimentation process.     

Probabilities of hitting a convex hull

In this note, we consider the non-negative least-square method with a random matrix. This problem has connections with the probability that the origin is not in the convex hull of many random points. As related problems, suitable estimates are obtained as well on the probability that a small ball does not hit the convex hull.