Nonlinear Optimization Line Search for Faster Derivative Approximation
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Solution Overview
Problem
Conventional nonlinear optimization methods require extensive calculations on functions and derivatives at each search step, leading to increased calculation time, especially in large-scale optimization problems, as they seek exact step sizes and direction vectors, which is not practical for reducing overall processing time.
Innovation Solution
A processing system that uses a line search method based on parabolic approximation to efficiently calculate step sizes and derivatives, approximating the first-order derivative at critical points using temporary step sizes and finite difference methods, reducing the number of calculations required at each search step.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional nonlinear optimization methods are used to calculate exact step sizes and direction vectors, then calculation precision is improved, but processing time increases significantly
Solution Approach 1:
The patent uses inexpensive approximation methods (parabolic approximation, finite difference methods) instead of expensive exact calculation methods. These approximation techniques provide sufficient accuracy for optimization while requiring significantly fewer computational operations, effectively replacing costly exact calculations with cheaper approximate ones that achieve the same practical goal.
Solution Approach 2:
The patent changes the calculation parameters from exact methods to approximate methods. Specifically, it uses parabolic approximation to estimate step sizes instead of exact line search, and finite difference methods to approximate derivatives instead of exact differentiation. This parameter change from exact to approximate calculations reduces computational complexity while maintaining adequate precision.
2Productivity
If the number of calculations on functions and derivatives is reduced, then processing speed is improved, but calculation accuracy deteriorates
Solution Approach 1:
The patent employs computationally inexpensive approximation techniques that require fewer calculations. Parabolic approximation uses only function value evaluations at a few points, and finite difference methods use simple arithmetic operations on these values, avoiding costly exact optimization subroutines while achieving sufficient accuracy for practical applications.
Solution Approach 2:
The patent replaces complex mechanical-like optimization procedures (exact line search, precise derivative calculations) with simpler mathematical approximation formulas. The parabolic approximation formula and finite difference formulas substitute for iterative exact calculation methods, reducing computational burden while maintaining functional equivalence for practical purposes.
Data Source
AI summary
A method of increasing the processing speed of a computer having a computer processing unit that executes a nonlinear optimization routine is described. A favorable approximation of the derivative f′(x+αd) at the critical point can be obtained by one differential calculation at each search step. As a result, when a large-scale nonlinear optimization problem requiring a large amount of calculations is processed, mk at each search step is reduced and consequently the calculation time T is reduced significantly.