Accelerated Optimization via Control Lyapunov Function
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Solution Overview
Problem
Conventional gradient descent-based methods for nonlinear optimization problems require a large number of iterations to reach an optimal point, making the optimization process time-consuming.
Innovation Solution
The implementation of an accelerated optimization method using a control Lyapunov function (CLF) and associated parameters, which generates a three-step iterative map to solve the optimization problem by advancing to new points and backtracking until a sufficient decrease in the objective function is achieved.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of operation
If conventional gradient descent-based methods are used for nonlinear optimization, then the optimization process is simple to implement, but the number of iterations required to reach an optimal point is large, making the process time-consuming
Solution Approach 1:
The patent applies preliminary action by performing multiple gradient calculations and evaluating multiple candidate points before committing to a direction of movement. Instead of immediately moving in the gradient direction, the method pre-calculates potential next steps and selects the most promising one, thereby reducing the total number of iterations needed to converge to an optimal solution while maintaining implementation simplicity
Solution Approach 2:
The patent implements dynamics by adapting the optimization step size and direction based on real-time evaluation of candidate points. The method dynamically adjusts the search strategy by calculating gradients at multiple points and selecting the direction that shows the most promise, rather than following a fixed gradient descent pattern, thus accelerating convergence without complicating the overall algorithm structure
2Loss of time
If the number of iterations is reduced to accelerate optimization, then the optimization time decreases, but the complexity of the optimization method increases
Solution Approach 1:
The patent applies segmentation by dividing the optimization step into multiple discrete sub-steps: calculating gradients at the current point, evaluating multiple candidate next points, selecting the best candidate, and updating the solution. This segmented approach allows the method to perform more thorough local analysis at each iteration, reducing the total number of iterations needed while keeping each individual step computationally manageable and the overall algorithm structure clear
3Use of energy by moving object
If conventional gradient descent is used, then the computational complexity per iteration is low, but many iterations are required to reach convergence
Solution Approach 1:
The patent applies preliminary action by pre-calculating gradients and evaluating multiple candidate next points before selecting the optimal direction. This preliminary analysis at each iteration allows the method to avoid wasted computations in subsequent iterations by choosing more promising search directions, thereby increasing convergence speed while the additional per-iteration cost is offset by the reduction in total iteration count
Data Source
AI summary
A method, apparatus and system including the execution of an accelerated optimization method. According to an exemplary embodiment, the accelerated optimization method includes: first-order optimality conditions for a generic nonlinear optimization problem are generated as part of the terminal transversality conditions of an optimal control problem. It is shown that the Lagrangian of the optimization problem is connected to the Hamiltonian of the optimal control problem via a zero-Hamiltonian, infinite-order, singular arc. The necessary conditions for the singular optimal control problem are used to produce an auxiliary controllable dynamical system whose trajectories generate algorithm primitives for the optimization problem. A three-step iterative map for a generic algorithm is designed by a semi-discretization step. Neither the feedback control law nor the differential equation governing the algorithm need be derived explicitly. A search direction is produced by a proximal-aiming-type method that dissipates a control Lyapunov function. New step size procedures based on minimizing control Lyapunov functions along a search vector complete the design of the accelerated optimization algorithms.