Numerical One-Dimensional Infinitesimal Analysis and Its Application to the Study of Optimization Problems

2019 ◽  
Author(s):  
Sergey Trofimov ◽  
Alexey Ivanov
Keyword(s):  
Optimization Problems ◽  
One Dimensional ◽  
Infinitesimal Analysis

scholarly journals Optimal Unimodular Sequences Design Method for Active Sensing Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Ze Li ◽  
Ping Li ◽  
Xinhong Hao ◽  
Xiaopeng Yan
Keyword(s):  
Optimization Problems ◽  
Design Method ◽  
Correlation Coefficients ◽  
Continuous Phase ◽  
Active Sensing ◽  
Sidelobe Level ◽  
One Dimensional ◽  
Sensing Systems ◽  
Practical Engineering ◽  
Dimensional Optimization

In active sensing systems, unimodular sequences with low autocorrelation sidelobes are widely adopted as modulation sequences to improve the distance resolution and antijamming performance. In this paper, in order to meet the requirements of specific practical engineering applications such as suppressing certain correlation coefficients and finite phase, we propose a new algorithm to design both continuous phase and finite phase unimodular sequences with a low periodic weighted integrated sidelobe level (WISL). With the help of the transformation matrix, such an algorithm decomposes the N-dimensional optimization problem into N one-dimensional optimization problems and then uses the iterative method to search the optimal solutions of the N one-dimensional optimization problems directly. Numerical experiments demonstrate the effectiveness and the convergence property of the proposed algorithm.

scholarly journals A bridging method for global optimization

1999 ◽  
Vol 41 (1) ◽  
pp. 41-57 ◽  
Author(s):  
Y. Liu ◽  
K. L. Teo
Keyword(s):  
Global Optimization ◽  
Differentiable Function ◽  
Numerical Solutions ◽  
Finite Interval ◽  
Optimization Problems ◽  
Optimal Solutions ◽  
Numerical Examples ◽  
One Dimensional ◽  
Differentiable Functions ◽  
Global Optimal

AbstractIn this paper a bridging method is introduced for numerical solutions of one-dimensional global optimization problems where a continuously differentiable function is to be minimized over a finite interval which can be given either explicitly or by constraints involving continuously differentiable functions. The concept of a bridged function is introduced. Some properties of the bridged function are given. On this basis, several bridging algorithm are developed for the computation of global optimal solutions. The algorithms are demonstrated by solving several numerical examples.

scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (1) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

Optimum Design of I. C. Engine Pistons

1984 ◽  
Vol 106 (2) ◽  
pp. 209-213 ◽  
Author(s):  
S. S. Rao ◽  
S. S. Srinivasa Rao
Keyword(s):  
Function Method ◽  
Optimization Problems ◽  
Interpolation Method ◽  
Unconstrained Minimization ◽  
Penalty Function Method ◽  
Minimum Volume ◽  
Constrained Optimization Problems ◽  
One Dimensional ◽  
Design Variables ◽  
Powell Method

The minimum volume design of I. C. engine pistons is considered with constraints on temperature and stresses developed in the piston. The interior penalty function method, coupled with the Davidon-Fletcher-Powell method of unconstrained minimization and the cubic interpolation method of one-dimensional search, is used for solving the constrained optimization problems. The temperature and stresses developed in the piston are determined by using the classic as well as the finite element methods of analysis. A sensitivity analysis is conducted to find the influence of changes in design variables on the objective function and the response parameters.

Optimal Supersonic Ejector Designs

1986 ◽  
Vol 108 (4) ◽  
pp. 414-420 ◽  
Author(s):  
J. C. Dutton ◽  
B. F. Carroll
Keyword(s):  
Flow Model ◽  
Optimization Problems ◽  
Pressure Ratio ◽  
Area Ratio ◽  
Stagnation Temperature ◽  
Entrainment Ratio ◽  
One Dimensional ◽  
Supersonic Ejector ◽  
Constant Area ◽  
The Common

A technique based on a one-dimensional constant area flow model has been developed for solving a large class of supersonic ejector optimization problems. In particular, the method determines the primary nozzle Mach number and ejector area ratio which optimizes either the entrainment ratio, compression ratio, or stagnation pressure ratio given values for the other two variables and the primary and secondary gas properties and stagnation temperatures. Design curves for the common case of diatomic primary and secondary gases of equal molecular weight and stagnation temperature are also presented and discussed.

scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (01) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

GEOMETRIC OPTIMIZATION PROBLEMS OVER SLIDING WINDOWS

2006 ◽  
Vol 16 (02n03) ◽  
pp. 145-157 ◽  
Author(s):  
TIMOTHY M. CHAN ◽  
BASHIR S. SADJAD
Keyword(s):  
Optimization Problems ◽  
Simple Algorithm ◽  
Sliding Window ◽  
Constant Factor ◽  
Geometric Optimization ◽  
One Dimension ◽  
Sliding Windows ◽  
One Dimensional ◽  
Previous Solution ◽  
Point Set

We study the problem of maintaining a (1 + ∊)-factor approximation of the diameter of a stream of points under the sliding window model. In one dimension, we give a simple algorithm that only needs to store [Formula: see text] points at any time, where the parameter R denotes the "spread" of the point set. This bound is optimal and improves Feigenbaum, Kannan, and Zhang's recent solution by two logarithmic factors. We then extend our one-dimensional algorithm to higher constant dimensions and, at the same time, correct an error in the previous solution. In high nonconstant dimensions, we also observe a constant-factor approximation algorithm that requires sublinear space. Related optimization problems, such as the width, are also considered in the two-dimensional case.

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