scholarly journals Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 283
Author(s):  
Tareq Hamadneh ◽  
Mohammed Ali ◽  
Hassan AL-Zoubi
Keyword(s):  
Lower Bounds ◽  
Linear Optimization ◽  
Rational Functions ◽  
Absolute Difference ◽  
Degree Elevation ◽  
Lower Bounding ◽  
Positive Polynomial ◽  
Affine Functions ◽  
The Given ◽  
Bernstein Coefficients

In this paper, we provide tight linear lower bounding functions for multivariate polynomials given over boxes. These functions are obtained by the expansion of polynomials into Bernstein basis and using the linear least squares function. Convergence properties for the absolute difference between the given polynomials and their lower bounds are shown with respect to raising the degree and the width of boxes and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate continuous rational function based on the Bernstein control points, the convex hull of a non-positive polynomial s, and degree elevation. Numerical comparisons with the well-known Bernstein constant lower bounding function are given. Finally, with these affine functions, the positivity of polynomials and rational functions can be certified by computing the Bernstein coefficients of their linear lower bounds.

scholarly journals Optimization of a k-covering of a bounded set with circles of two given radii

2021 ◽  
Vol 11 (1) ◽  
pp. 232-240
Author(s):  
Alexander V. Khorkov ◽  
Shamil I. Galiev
Keyword(s):  
Linear Programming ◽  
Numerical Method ◽  
Lower Bounds ◽  
Integer Linear Programming ◽  
Numerical Results ◽  
Programming Models ◽  
Bounded Set ◽  
The Given

Abstract A numerical method for investigating k-coverings of a convex bounded set with circles of two given radii is proposed. Cases with constraints on the distances between the covering circle centers are considered. An algorithm for finding an approximate number of such circles and the arrangement of their centers is described. For certain specific cases, approximate lower bounds of the density of the k-covering of the given domain are found. We use either 0–1 linear programming or general integer linear programming models. Numerical results demonstrating the effectiveness of the proposed methods are presented.

Lower Bounds to the Critical Rayleigh Number in Completely Confined Regions

1967 ◽  
Vol 34 (2) ◽  
pp. 308-312 ◽  
Author(s):  
M. Sherman ◽  
S. Ostrach
Keyword(s):  
Rayleigh Number ◽  
Lower Bounds ◽  
Body Force ◽  
Critical Rayleigh Number ◽  
Convective Motion ◽  
The Body ◽  
Maximum Dimension ◽  
Fixed Temperature ◽  
Boussinesq Fluid ◽  
The Given

A method is presented for estimating lower bounds to the minimum Rayleigh number that will induce a state of convective motion in a quasi-incompressible (Boussinesq) fluid where the temperature gradient is in the direction of the body force. The fluid is completely confined by fixed-temperature, rigid bounding walls. For any arbitrary region, the critical Rayleigh number is greater than 1558(h/D)4 where h is the maximum dimension of the given region in the direction of the body force and D is the diameter of an equal volume sphere. In certain simple geometrical configurations, improved lower-bound estimates are calculated.

scholarly journals On Gap-Based Lower Bounding Techniques for Best-Arm Identification

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 788
Author(s):  
Lan V. Truong ◽  
Jonathan Scarlett
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Sample Complexity ◽  
Bandit Problem ◽  
Lower Bounding ◽  
Complexity Lower Bounds

In this paper, we consider techniques for establishing lower bounds on the number of arm pulls for best-arm identification in the multi-armed bandit problem. While a recent divergence-based approach was shown to provide improvements over an older gap-based approach, we show that the latter can be refined to match the former (up to constant factors) in many cases of interest under Bernoulli rewards, including the case that the rewards are bounded away from zero and one. Together with existing upper bounds, this indicates that the divergence-based and gap-based approaches are both effective for establishing sample complexity lower bounds for best-arm identification.

Matrix Perturbation and Optimal Partition Invariancy in Linear Optimization

2015 ◽  
Vol 32 (03) ◽  
pp. 1550013 ◽  
Author(s):  
Alireza Ghaffari-Hadigheh ◽  
Nayyer Mehanfar
Keyword(s):  
Linear Optimization ◽  
Optimal Solution ◽  
Singular Values ◽  
Coefficient Matrix ◽  
Optimal Partition ◽  
Matrix Perturbation ◽  
Optimal Value ◽  
Special Perturbation ◽  
Transition Points ◽  
The Given

Understanding the effect of variation of the coefficient matrix in linear optimization problem on the optimal solution and the optimal value function has its own importance in practice. However, most of the published results are on the effect of this variation when the current optimal solution is a basic one. There is only a study of the problem for special perturbation on the coefficient matrix, when the given optimal solution is strictly complementary and the optimal partition (in some sense) is known. Here, we consider an arbitrary direction for perturbation of the coefficient matrix and present an effective method based on generalized inverse and singular values to detect invariancy intervals and corresponding transition points.

scholarly journals An interior point algorithm for solving linear optimization problems using a new trigonometric kernel function

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1471-1486
Author(s):  
S. Fathi-Hafshejani ◽  
Reza Peyghami
Keyword(s):  
Kernel Function ◽  
Interior Point ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Interior Point Algorithm ◽  
Worst Case ◽  
Complexity Bounds ◽  
Linear Optimization Problems ◽  
Primal Dual ◽  
The Given

In this paper, a primal-dual interior point algorithm for solving linear optimization problems based on a new kernel function with a trigonometric barrier term which is not only used for determining the search directions but also for measuring the distance between the given iterate and the ?-center for the algorithm is proposed. Using some simple analysis tools and prove that our algorithm based on the new proposed trigonometric kernel function meets O (?n log n log n/?) and O (?n log n/?) as the worst case complexity bounds for large and small-update methods. Finally, some numerical results of performing our algorithm are presented.

scholarly journals Теорема Артина для f-колец

2015 ◽  
Author(s):  
A.G. Kusraev
Keyword(s):  
Rational Functions ◽  
Sum Of Squares ◽  
Famous Theorem ◽  
The Real ◽  
Complete Ring ◽  
Positive Polynomial ◽  
Ring Of Quotients ◽  
Real Closure ◽  
F Ring ◽  
17Th Problem

The main result states that each positive polynomial p in N variables with coefficients in a unital Archimedean f-ring K is representable as a sum of squares of rational functions over the complete ring of quotients of K provided that p is positive on the real closure of K. This is proved by means of Boolean valued interpretation of Artin's famous theorem which answers Hilbert's 17th problem affirmatively.

Das ideale Plentergleichgewicht – Leitbild oder Luxus? (Essay) | The ideal equilibrium state in a selection forest – vision or luxury? (essay)

2008 ◽  
Vol 159 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Dana Sonnemann
Keyword(s):  
Stand Structure ◽  
Linear Time ◽  
Linear Optimization ◽  
Maximum Diameter ◽  
Growing Stock ◽  
Management Objectives ◽  
Management Interventions ◽  
Linear Optimization Model ◽  
Selection Forest ◽  
The Given

In managing unevenly aged forests, conventional ideal equilibrium models based on curves of declining stem numbers have relied on the overall concept of a simultaneous fulfilment of various aims. The study presented here investigated whether such models are adaptable to current economic conditions and to what extent financial objectives could be optimized without infringing on structural requirements. A linear, time discrete equilibrium model calibrated with data collected from a sample area within a Swiss selection forest was combined with a linear optimization model. Financial objectives were optimized in numeric experiments under a range of strictness values for stand-structure constraints. The results were compared with those obtained from an exemplary conservative ideal model. This evaluation found system equilibria far from that reference, and it also found some wich could generate a much higher income. However, the most meaningful increase of the target variable could be realized without requiring any critical deviation from the compared model. This could be accomplished by augmenting the current growing stock while reducing the maximum diameter at breast height. Therefore, management interventions that incur losses could be abandoned in the lower diameter classes, and only trees above a specified diameter would then be harvested. In applying this new model, the decision makers would be asked to thoroughly analyze their management objectives and the given restrictions on acting accordingly. Hence, they could immediately provide arguments to justify their decisions.

LOWER BOUNDS FOR THE BINDING ENERGIES OF N-PARTICLE SYSTEMS

1967 ◽  
Vol 45 (11) ◽  
pp. 3753-3760 ◽  
Author(s):  
S. Pruski
Keyword(s):  
Lower Bounds ◽  
State Energy ◽  
Nuclear Charge ◽  
Energy Levels ◽  
Binding Energies ◽  
Particle Systems ◽  
Fermion System ◽  
Close Approximation ◽  
Bounded Below ◽  
The Given

Two new families of lower bounds to the energy of an N-fermion system interacting via two-body forces are derived. They involve, respectively, energy levels of (N − 1)- and (N − 2)-particle systems associated with the given N-fermion system. The first is expected to give a fairly close approximation to the energy. In the case of an atom of nuclear charge Z with N electrons, the ground-state energy of this [Z, N] ion is bounded below by (N − 1)/(N − 2)2 times the sum of the N lowest energy levels of an associated ion [Z′, N′], with N′ = N − 1 and Z′ = Z(N − 2)/(N − 1).

scholarly journals The hardest halfspace

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Alexander A. Sherstov
Keyword(s):  
Lower Bounds ◽  
Rational Approximation ◽  
Communication Complexity ◽  
Rational Functions ◽  
Explicit Construction ◽  
Upper Bounds ◽  
Infinity Norm ◽  
Communication Problem

AbstractWe study the approximation of halfspaces $$h:\{0,1\}^n\to\{0,1\}$$ h : { 0 , 1 } n → { 0 , 1 } in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the “hardest” halfspace, for which we prove polynomial and rational approximation lower bounds that match the trivial upper bounds achievable for all halfspaces. This completes a lengthy line of work started by Myhill and Kautz (1961). As an application, we construct a communication problem that achieves essentially the largest possible separation, of O(n) versus $$2^{-\Omega(n)}$$ 2 - Ω ( n ) , between the sign-rank and discrepancy. Equivalently, our problem exhibits a gap of log n versus $$\Omega(n)$$ Ω ( n ) between the communication complexity with unbounded versus weakly unbounded error, improving quadratically on previous constructions and completing a line of work started by Babai, Frankl, and Simon (FOCS 1986). Our results further generalize to the k-party number-on-the-forehead model, where we obtain an explicit separation of log n versus $$\Omega(n/4^{n})$$ Ω ( n / 4 n ) for communication with unbounded versus weakly unbounded error.

scholarly journals Statistical Query Algorithms for Mean Vector Estimation and Stochastic Convex Optimization

2021 ◽  
Author(s):  
Vitaly Feldman ◽  
Cristóbal Guzmán ◽  
Santosh Vempala
Keyword(s):  
Machine Learning ◽  
Convex Optimization ◽  
Lower Bounds ◽  
Differential Privacy ◽  
Linear Optimization ◽  
General Setting ◽  
Upper And Lower Bounds ◽  
Statistical Queries ◽  
The Mean ◽  
Mean Vector

Stochastic convex optimization, by which the objective is the expectation of a random convex function, is an important and widely used method with numerous applications in machine learning, statistics, operations research, and other areas. We study the complexity of stochastic convex optimization given only statistical query (SQ) access to the objective function. We show that well-known and popular first-order iterative methods can be implemented using only statistical queries. For many cases of interest, we derive nearly matching upper and lower bounds on the estimation (sample) complexity, including linear optimization in the most general setting. We then present several consequences for machine learning, differential privacy, and proving concrete lower bounds on the power of convex optimization–based methods. The key ingredient of our work is SQ algorithms and lower bounds for estimating the mean vector of a distribution over vectors supported on a convex body in Rd. This natural problem has not been previously studied, and we show that our solutions can be used to get substantially improved SQ versions of Perceptron and other online algorithms for learning halfspaces.

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