Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth...

Area of Science:

• Optimization Theory
• Convex Analysis
• Numerical Methods

Background:

• Minimizing convex functions is a fundamental problem in optimization.
• Existing methods may face challenges with complex constraint sets.
• The set of minima of a differentiable convex function forms a convex set.

Purpose of the Study:

• To develop a novel algorithm for minimizing a smooth convex objective function.
• To address constraints defined by the set of minima of another convex function.
• To analyze the convergence properties of the proposed method.

Main Methods:

• A hybrid approach combining gradient descent with a penalization technique.
• Incorporation of an inertial term to leverage the history of iterates.
• Convergence analysis utilizing Opial Lemma and generalized Fejér monotonicity.

Main Results:

• Weak convergence of the generated iterates to an optimal solution is proven.
• Convergence of objective function values to the optimal value is established.
• A condition for convergence is derived using the Fenchel conjugate of the constraint function.

Conclusions:

• The proposed algorithm effectively solves the constrained convex optimization problem.
• The inertial term enhances the convergence properties of the method.
• Numerical experiments, including support vector machine-based image classification, validate the algorithm's functionality.