Gradient Methods on Strongly Convex Feasible Sets and Optimal Control of Affine Systems.

Area of Science:

• Optimization Theory
• Numerical Analysis
• Optimal Control

Background:

• Gradient projection and conditional gradient methods are key algorithms for minimization problems.
• Existing methods often require convex objective functions, limiting their applicability.
• New discretization techniques for optimal control problems introduce strongly convex constraints but may have non-convex objective functionals.

Purpose of the Study:

• To analyze the convergence properties of gradient projection and conditional gradient methods for abstract minimization problems.
• To investigate convergence on strongly convex sets, even when the objective functional is not convex.
• To demonstrate the applicability of these methods to optimal control problems solved with a novel discretization technique.

Main Methods:

• Theoretical analysis of convergence for gradient projection and conditional gradient algorithms.
• Application of abstract results to specific cases, including linear-quadratic affine optimal control problems.
• Numerical simulations to validate the theoretical findings.

Main Results:

• Linear convergence is proven for the studied methods on strongly convex sets, irrespective of objective functional convexity.
• The developed discretization technique yields higher accuracy and involves strongly convex constraints.
• The abstract convergence results are successfully applied to linear-quadratic affine optimal control problems.

Conclusions:

• The gradient projection and conditional gradient methods exhibit robust linear convergence under specific conditions (strongly convex sets).
• These methods are effective for optimal control problems utilizing advanced discretization techniques, even with non-convex objective functionals.
• Numerical evidence supports the theoretical guarantees of linear convergence.