A Scalable Algorithm for Sparse Portfolio Selection

The sparse portfolio selection problem is one of the most famous and frequently studied problems in the optimization and financial economics literatures. In a universe of risky assets, the goal is to construct a portfolio with maximal expected return and minimum variance, subject to an upper bound on the number of positions, linear inequalities, and minimum investment constraints. Existing certifiably optimal approaches to this problem have not been shown to converge within a practical amount of time at real-world problem sizes with more than 400 securities. In this paper, we propose a more scalable approach. By imposing a ridge regularization term, we reformulate the problem as a convex binary optimization problem, which is solvable via an efficient outer-approximation procedure. We propose various techniques for improving the performance of the procedure, including a heuristic that supplies high-quality warm-starts, and a second heuristic for generating additional cuts that strengthens the root relaxation. We also study the problem’s continuous relaxation, establish that it is second-order cone representable, and supply a sufficient condition for its tightness. In numerical experiments, we establish that a conjunction of the imposition of ridge regularization and the use of the outer-approximation procedure gives rise to dramatic speedups for sparse portfolio selection problems.

Discrete Time Multi-servers Loss Systems and Stochastic Approximation with Delayed Groups of Customers

The stochastic approximation procedure with delayed groups of delayed customers is investigated. The Robbins-Monro stochastic approximation procedure is adjusted to be usable in the presence of delayed groups of delayed customers. Two loss systems are introduced to get an accurate description of the proposed procedure. Each customer comes after fixed time-intervals with the stage of the following customer is accurate according to the outcome of the preceding one, where the serving time of a customer is assumed to be discrete random variable. Some applications of the procedure are given where the analysis of their results is obtained. The analysis shows that efficiencies of the procedure can be increased by minimizing the number of customers of a group irrespective of their service times that may take maximum values. Efficiencies depend on the maximum service time of the customer and on the number of customers of the group. The most important result is that efficiencies of the procedure are increased by increasing the service time distributions as well as service times of customers .This new situation can be applied to increase the number of served customers where the number of served groups will also be increased. The results obtained seem to be acceptable. In general, our proposal can be utilized to other stochastic approximation procedures to increase the production in many fields such as medicine, computer sciences, industry, and applied sciences.

Approximation of Non-Linear Rotor Dynamic Resonance Behavior of Vertically Aligned Hydro-Units Guided by Tilting-Pad Bearings

Dynamic analyses of vertical hydro power plant rotors require the consideration of the non-linear bearing characteristics. This study investigates the vibrational behavior of a typical vertical machine using a time integration method that considers non-linear bearing forces. Thereby, the influence of support stiffness and unbalance magnitude is examined. The results show a rising influence of unbalance on resonance speed with increasing support stiffness. Furthermore, simulations reveal that the shaft orbit in the bearing is nearly circular for typical design constellations. This property is applied to derive a novel approximation procedure enabling the examination of non-linear resonance behavior, using linear rotor dynamic theory. The procedure considers the dynamic film pressure for determining the pad position. In addition, it is time-efficient compared to a time integration method, especially at high amplitudes when damping becomes small.

Approximation properties of a new family of Gamma operators and their applications

The present paper introduces a new modification of Gamma operators that protects polynomials in the sense of the Bohman–Korovkin theorem. In order to examine their approximation behaviours, the approximation properties of the newly introduced operators such as Voronovskaya-type theorems, rate of convergence, weighted approximation, and pointwise estimates are presented. Finally, we present some numerical examples to verify that the newly constructed operators are an approximation procedure.

On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space

PurposeIn the present paper, the authors will discuss the solvability of a class of nonlinear anisotropic elliptic problems (P), with the presence of a lower-order term and a non-polynomial growth which does not satisfy any sign condition which is described by an N-uplet of N-functions satisfying the Δ2-condition, within the fulfilling of anisotropic Sobolev-Orlicz space. In addition, the resulting analysis requires the development of some new aspects of the theory in this field. The source term is merely integrable.Design/methodology/approachAn approximation procedure and some priori estimates are used to solve the problem.FindingsThe authors prove the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain. The resulting analysis requires the development of some new aspects of the theory in this field.Originality/valueTo the best of the authors’ knowledge, this is the first paper that investigates the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain.

Epsilon Best Arm Identification in Spectral Bandits

We propose an analysis of Probably Approximately Correct (PAC) identification of an ϵ-best arm in graph bandit models with Gaussian distributions. We consider finite but potentially very large bandit models where the set of arms is endowed with a graph structure, and we assume that the arms' expectations μ are smooth with respect to this graph. Our goal is to identify an arm whose expectation is at most ϵ below the largest of all means. We focus on the fixed-confidence setting: given a risk parameter δ, we consider sequential strategies that yield an ϵ-optimal arm with probability at least 1-δ. All such strategies use at least T*(μ)log(1/δ) samples, where R is the smoothness parameter. We identify the complexity term T*(μ) as the solution of a min-max problem for which we give a game-theoretic analysis and an approximation procedure. This procedure is the key element required by the asymptotically optimal Track-and-Stop strategy.

Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

We prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.

Computing the Sound of the Sea in a Seashell

The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $${{\mathcal {C}}}^2$$ C 2 . The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.

Approximation Procedure for Determining Oscillation Period of Frame Structure

Dynamic analysis can be used to find dynamic displacements, time history, and the frequency content of the load. One analysis technique for calculating the linear response of structures to dynamic loading is a modal analysis. In modal analysis, we decompose the response of the structure into several vibration modes. A mode is defined by its frequency and shape. Structural engineers call the mode with the shortest frequency (the longest period) the fundamental mode. Holzer and Stodola's approximate methods for determining the forms and periods of oscillation for frame structures are presented in the paper. An approximation method, based on approximate relative stiffnesses of the storeys and the ground floor, is analyzed and proposed. The results obtained by the proposed approximate procedure do not greatly deviate from those obtained by more accurate calculations. It is therefore emphasized that the method can be used both in practice and for checking computer-based analysis of complex systems. At the end of the paper was given a comparison of the results obtained by approximate methods and some engineering software.

Control problem for the impulse process under stochastic optimization procedure and Levy conditions

A stochastic approximation procedure and a limit generator of the original problem are constructed for a system of stochastic differential equations with Markov switching and impulse perturbation under Levy approximation conditions with control, which is determined by the condition for the extremum of the quality criterion function.The control problem using the stochastic optimization procedure is a generalization of the control problem with the stochastic approximation procedure, which was studied in previous works of the authors. This generalization is not simple and requires non-trivial approaches to solving the problem. In particular we discuss how the behavior of the boundary process depends on the prelimiting stochastic evolutionary system in the ergodic Markov environment. The main assumption is the condition for uniform ergodicity of the Markov switching process, that is, the existence of a stationary distribution for the switching process over large time intervals. This allows one to construct explicit algorithms for the analysis of the asymptotic behavior of a controlled process. An important property of the generator of the Markov switching process is that the space in which it is defined splits into the direct sum of its zero-subspace and a subspace of values, followed by the introduction of a projector that acts on the subspace of zeros.For the first time, a model of the control problem for the diffusion transfer process using the stochastic optimization procedure for control problem is proposed. A singular expansion in the small parameter of the generator of the three-component Markov process is obtained, and the problem of a singular perturbation with the representation of the limiting generator of this process is solved.