The problem of fitting ellipsoids occurs in many areas of science. It is useful in pattern recognition, particle physics, computer graphics, medical imaging of organs, and statistical error analysis. We describe an algorithm for finding an equation for a multi-dimensional ellipsoid fit to a distribution of discrete data points. In general, we find that the algorithm works well for fitting ellip...

R d is log-concave if p = e where φ :Rd → [−∞,∞) is concave. We denote the class of all such densities p on R by Pd,0. Log-concave densities are always unimodal and have convex level sets. Furthermore, log-concavity is preserved under marginalization and convolution. Thus, the classes of log-concave densities can be viewed as natural nonparametric extensions of the class of Gaussian densities. ...

This paper proposes a method based on linear programming techniques to treat quasi-concave and non-concave fuzzy multi-objective programming (FMOP) problems. The proposed method initially presents a piecewise linear expression to interpreting a quasi-concave membership function. Then we 7nd the convex-type break points and transform all quasiconcave membership functions into concave functions. ...

A function F defined on all subsets of a finite ground set E is quasiconcave if F (X∪Y ) ≥ min{F (X), F (Y )} for all X, Y ⊂ E. Quasi-concave functions arise in many fields of mathematics and computer science such as social choice, theory of graph, data mining, clustering and other fields. The maximization of quasi-concave function takes, in general, exponential time. However, if a quasi-concav...

inf u 6 Ttu(x) 6 u(x) 6 Ťtu(x) 6 supu for each t > 0 and each x ∈ H. A function u : H −→ R is called k-semi-concave, k > 0, if the function x −→ u(x)−‖x‖2/k is concave. The function u is called k-semi-convex if −u is k-semiconcave. A bounded function u is t-semi-concave if and only if it belongs to the image of the operator Tt, this follows from Lemma 1 and Lemma 3 below. A function is called s...

Improvement of light extraction efficiency of InGaN light emitting diodes (LEDs) using polydimethylsiloxane (PDMS) concave microstructures arrays was demonstrated. The size effect of the concave microstructures on the light extraction efficiency of III-Nitride LEDs was studied. Depending on the size of the concave microstructures, ray tracing simulations show that the use of PDMS concave micros...

A class of probabilistic constrained programming problems are considered where the probabilistic constraint is of the form P{gi(x, ξ) ≥ 0, i = 1, . . . , r} ≥ p and the functions gi, i = 1, . . . , r are concave. It is shown that the x-function on the left hand side is logarithmic concave provided ξ has a logarithmic concave density. Special cases are mentioned and algorithmic solution of probl...

Linear optimization (LO) is the fundamental problem of mathematical optimization. It admits an enormous number of applications in economics, engineering, science and many other fields. The three most significant classes of algorithms for solving LO problems are: Pivot, Ellipsoid and Interior Point Methods. Because Ellipsoid Methods are not efficient in practice we will concentrate on the comput...

0 < λ ≤ f ≤ Λ in Ω, and for some x ∈ Ω, Sh(x) ⊂⊂ Ω, then Sh(x) is equivalent to an ellipsoid centered at x i.e. kE ⊂ Sh(x)− x ⊂ k−1E for some ellipsoid E of volume h and for a constant k > 0 which depends only on λ,Λ, n. This property provides compactness of sections modulo affine transformations. This is particularly useful when dealing with interior C and W 2,p estimates of strictly convex so...