Gradient Projection Method with Constraints

The Gradient Projection Method with Constraints is a numerical computation technique designed for solving constrained optimization problems. It combines the principles of gradient descent with projection operations to ensure that solutions remain within the feasible region throughout the iterative process.

The fundamental mechanism involves calculating the gradient direction of the objective function during each iteration, followed by searching along this direction. When constraint boundaries are encountered, the current point is projected back into the feasible domain. This projection operation maintains feasibility while leveraging gradient information for optimization.

In MATLAB implementation, several critical steps require attention: The first step involves designing a projection operator for the feasible region, which must be adapted to specific constraint types. For linear constraints, the projection operator can be solved using quadratic programming techniques, while for bound constraints, variable values exceeding boundaries can be directly truncated through element-wise clipping operations.

The second step requires appropriate step size selection strategies. Options include fixed step sizes or adaptive adjustments like Armijo line search, which ensures convergence by satisfying sufficient decrease conditions. Empirical testing shows adaptive step sizes generally yield superior performance.

Termination criteria typically involve gradient norms falling below a specified threshold or objective function value changes smaller than a tolerance limit. Practical implementations should incorporate maximum iteration limits to prevent infinite loops in non-convergent cases.

Verified MATLAB implementations demonstrate that this method performs effectively for small to medium-scale constrained optimization problems, particularly with simple constraints where convergence speed and solution accuracy are well-balanced. However, for highly non-convex problems or complex constraints, hybrid approaches combining additional techniques may be necessary to enhance performance.