The least concave majorant, $\hat F$, of a continuous function $F$ on a closed interval, $I$, is defined by $ \hat F (x) = \inf\{ G(x) G \geq F, G \text{ concave}\},\quad x \in I. $ We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$. Given any function $F \in\mathcal{C}^4(I)$, it can be well-approximated on $I$ by a clamped cubic spline $S$. We show that $\hat S$ is then a good approximation to $\hat F$. We give two examples, one to illustrate, the other to apply our algorithm., Martin Franců, Ron Kerman, Gord Sinnamon., and Obsahuje bibliografii
V práci je prezentovaná moderná paradigma darvinovskej evolúcie, ktorá ju chápe ako univerzálny optimalizačný algoritmus, aplikovateľný nielen v biologických vedách, ale aj v roznych oblastiach tak exaktných, ako aj sociálnych a behaviorálnych vied., Vladimír Kvasnička, Jiří Pospíchal., and Obsahuje bibliografii
In this report we propose a new recursive matrix formulation of limited memory variable metric methods. This approach can be used for an arbitrary update from the Broyden class (and some other updates) and also for the approximation of both the Hessian matrix and its inverse. The new recursive formulation requires approximately 4mn multiplications and additions per iteration, so it is comparable with other efficient limited memory variable metric methods. Numerical experiments concerning Algorithm 1, proposed in this report, confirm its practical efficiency.
The purpose of the paper is to present existing and discuss modified optimization models and solution techniques which are suitable for engineering decision-making problems containing random elements with emphasis on two decision stages. The developed aproach is called two-stage stochastic programming and the paper links motivation, applicability, theoretical remarks, transformations, input data generation techniques, and selected decomposition algorithms for generalized class of engineering problems. The considered techniques have been found applicable by the experience of the authors in various areas of engineering problems. They have been applied to engineering design problems involving constraints based on differential equations to achieve reliable solutions. They have served for technological process control e.g. in melting, casting, and sustainable energy production. They have been used for industrial production technologies involving related logistics, as e.g. fixed interval scheduling under uncertainty. The paper originally introduces several recent improvements in the linked parts and it focuses on the unified two-stage stochastic programming approach to engineering problems in general. It utilizes authentic experience and ideas obtained in certain application areas and advises their fruitful utilization for other cases. The paper follows the paper published in 2000 which deals with the applicability of static stochastic programs to engineering design problems. Therefore, it refers to the basic concepts and notation introduced there and reviews only the principal ideas in the beginning. Then. it focuses on motivation of recourse concepts and two decision stages from engineering point of view. The principal models are introduced and selected theoretical features are reviewed. They are also accompanied by the discussion about difficulties caused by real-world cases. Scenario-based approach is detailed as the important one for the solution of engineering problems, discussion in data input generation is added together with model transformation remarks. Robust algorithms suitable for engineering problems involving nonlinearities and integer variable are selected and scenario-based decomposition is preferred. An original experience with using heuristics is shared. Several postprocessing remarks are added at the end of the paper, which is followed by an extensive literature review. and Obsahuje seznam literatury
A universal set of equations for determining chlorophyll (Chl) a, accessory Chl b, c, and d, and total Chl have been developed for 90 % acetone, 100 % methanol, and ethanol solvents suitable for estimating Chl in extracts from natural assemblages of algae. The presence of phaeophytin (Ph) a not only interferes with estimates of Chl a but also with Chl b and c determinations. The universal algorithms can hence be misleading if used on natural collections containing large amounts of Ph. The methanol algorithms are severely affected by the presence of Ph and so are not recommended. The algorithms were tested on representative mixtures of Chls prepared from extracts of algae with known Chl composition. The limits of detection (and inherent error, ±95 % confidence limit) for all the Chl equations were less than 0.03 g m-3. The algorithms are both accurate and precise for Chl a and d but less accurate for Chl b and c. With caution the algorithms can be used to calculate a Chl profile of natural assemblages of algae. The relative error of measurements of Chls increases hyperbolically in diluted extracts. For safety reasons, efficient extraction of Chls and the convenience of being able to use polystyrene cuvettes, the algorithms for ethanol are recommended for routine assays of Chls in natural assemblages of aquatic plants.