Gradient Projection Method with Constraints

The Gradient Projection Method with Constraints is an efficient numerical approach for solving constrained optimization problems. When searching for extremum values of objective functions under specific constraints, this method ensures iterative processes always satisfy constraints by projecting search directions onto feasible region boundaries.

The core algorithmic concept involves calculating the objective function's gradient at each iteration step and projecting it onto the tangent space of the current feasible region. This projection operation guarantees that search directions move along constraint boundaries, preventing constraint violations. For linear constraint problems, projection matrices can be directly computed; for nonlinear constraints, linear approximation methods are required.

In MATLAB implementation, key steps include: initializing feasible points, computing gradient vectors, constructing projection matrices, determining step-size parameters, and convergence verification. The algorithm dynamically adjusts search directions based on current point positions - when approaching constraint boundaries, it automatically modifies directions to maintain feasibility. Implementation typically involves functions like `gradient()` for derivative calculation and matrix operations for projection computations.

Practical applications require careful handling of projection matrix orthogonality and selection of line search strategies like Armijo rule. This method is particularly suitable for medium-scale convex optimization problems with well-defined constraints, demonstrating extensive application value in engineering optimization and economic modeling domains.