ADMM Step Size Optimization for Stochastic Quadratic Programs
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Solution Overview
Problem
Current methods for solving large-scale stochastic quadratic programs in model predictive control are hindered by high computational complexity per iteration, slow convergence, and difficulty in handling various constraints, making them unsuitable for real-time applications in systems with uncertain parameters.
Innovation Solution
The method employs an Alternating Direction Method of Multipliers (ADMM) with optimized step sizes and conjugate gradient acceleration to solve stochastic quadratic programs with linear constraints, decomposing problems into simpler blocks and using projection onto convex sets to reduce computational complexity and enhance convergence.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional methods (active set algorithm, interior point algorithm) are used to solve MPC generated QP problems, then solution accuracy is maintained, but computational effort per iteration becomes prohibitive for large scale problems
Solution Approach 1:
The patent segments the QP problem into two separate subproblems: an equality-constrained QP and a projection onto a convex set. This decomposition allows each subproblem to be solved more efficiently using specialized algorithms, reducing the overall computational burden while maintaining solution accuracy. The segmentation transforms a single complex optimization problem into two simpler, more tractable problems that can be solved alternately.
2Productivity
If fast gradient algorithm or fast optimization methods are used for MPC, then computational complexity is reduced, but ability to handle linear inequality constraints is limited
Solution Approach 1:
The patent introduces an intermediary variable and transforms the constrained optimization problem into an augmented Lagrangian formulation. This intermediary approach allows the use of fast gradient methods while still handling linear inequality constraints through the augmented Lagrangian terms, which act as mediators between the objective function and the constraints.
3Reliability
If ADMM is used to solve stochastic quadratic programs, then convergence is achieved, but step size selection is complex and convergence speed is slow
Solution Approach 1:
The patent changes the parameter selection strategy by deriving optimal step sizes based on the eigenvalues of the Hessian matrix. This parameter optimization transforms the standard ADMM into an accelerated version that maintains convergence guarantees while significantly reducing the number of iterations required. The step size is explicitly formulated as a function of problem-specific parameters rather than using fixed or heuristic values.
Data Source
AI summary
A method solves a stochastic quadratic program (StQP) for a convex set with a set of general linear equalities and inequalities by an alternating direction method of multipliers (ADMM). The method determines an optimal solution, or certifies that no solution exists. The method optimizes a step size β for the ADMM. The method is accelerated using a conjugate gradient (CG) method. The StMPC problem is decomposed into two blocks. The first block corresponds to an equality constrained QP, and the second block corresponds to a projection onto the StMPC inequalities and anticipativity constraints. The StMPC problem can be decomposed into a set of time step problems, and then iterated between the time step problems to solve the decoupled problems until convergence.