8.4 Applications to optimization and variational inequalities

Optimization and variational inequalities are key applications of monotone operators. This section shows how the proximal point algorithm can solve these problems by reformulating them as monotone inclusions.

We'll explore how the algorithm generates convergent sequences for convex optimization and variational inequalities. We'll also discuss the strengths and limitations of proximal point methods in tackling these problems.

Proximal Point Algorithm for Optimization

Iterative Method and Convergence

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• The proximal point algorithm is an iterative method for solving convex optimization problems by solving a sequence of regularized subproblems
• The algorithm generates a sequence of points that converge to the optimal solution of the original problem under certain conditions
• The convergence rate of the proximal point algorithm depends on the properties of the objective function, such as strong convexity or Lipschitz continuity of the gradient

Subproblems and Solving Techniques

• Each subproblem involves minimizing the sum of the original objective function and a quadratic proximal term, which encourages the next iterate to be close to the current one
• The proximal term is weighted by a positive parameter that controls the strength of the regularization and the step size of the algorithm
• The subproblems can be solved efficiently using various methods, such as gradient descent or Newton's method (for smooth objective functions) or proximal gradient method (for non-smooth objective functions), depending on the structure of the objective function
• The algorithm requires the choice of suitable stopping criteria, such as the relative change in the objective value or the norm of the gradient, to terminate the iterations

Variational Inequalities as Monotone Inclusions

Reformulation and Monotone Operators

• Variational inequality problems can be reformulated as monotone inclusion problems, which involve finding a point in the intersection of the graphs of two monotone operators
• A variational inequality problem is defined by a set and a mapping, and it seeks a point in the set such that the mapping at this point forms an acute angle with any other point in the set
• The mapping in the variational inequality problem is usually a monotone operator, which means that it satisfies a certain inequality relating the inner product of the difference between two points and the difference between their images under the mapping

Normal Cone Operator and Monotone Inclusion

• The set in the variational inequality problem is typically a closed and convex subset of a Hilbert space, such as the non-negative orthant (for non-negativity constraints) or a polyhedron (for linear constraints)
• The reformulation of a variational inequality problem as a monotone inclusion problem involves defining two monotone operators: one is the normal cone operator of the set, and the other is the mapping itself
• The normal cone operator associates each point in the set with the set of all vectors that form an obtuse angle with any vector pointing from that point to another point in the set
• The monotone inclusion problem seeks a point at which the sum of the two monotone operators contains the zero vector, which is equivalent to the variational inequality problem

Solving Variational Inequalities with Proximal Points

Adapted Algorithm and Convergence

• The proximal point algorithm can be adapted to solve variational inequality problems by exploiting their reformulation as monotone inclusion problems
• The algorithm generates a sequence of points that converge to a solution of the variational inequality problem under certain conditions on the set and the mapping
• The algorithm can handle variational inequality problems with multiple solutions by converging to one of them, depending on the initialization and the regularization parameter

Regularized Subproblems and Solving Methods

• Each iteration of the algorithm involves solving a regularized subproblem that consists of finding a point that satisfies a perturbed version of the variational inequality
• The perturbation is introduced by adding a proximal term to the mapping, which encourages the next iterate to be close to the current one and ensures the existence and uniqueness of the solution to the subproblem
• The subproblems can be solved using various methods, such as projection methods (for simple constraints) or splitting algorithms (for composite mappings), depending on the structure of the set and the mapping
• The choice of the regularization parameter in the proximal term affects the convergence speed and the numerical stability of the algorithm

Advantages vs Limitations of Proximal Point Methods

Strengths and Benefits

• Proximal point methods have several advantages in solving optimization and variational problems, such as their simplicity, flexibility, and robustness
• They can handle a wide range of problems, including those with non-smooth or non-differentiable objective functions (such as l1-norm regularization), constraints (such as bound constraints), or operators (such as subdifferentials)
• They enjoy global convergence properties under mild assumptions, such as the monotonicity or the Lipschitz continuity of the operators involved
• They can be implemented using various algorithms for solving the subproblems, which can be chosen based on the structure and the properties of the problem at hand
• They can be accelerated using techniques such as extrapolation (Nesterov's acceleration), inertia (heavy-ball method), or variable metric (quasi-Newton methods), which can improve their convergence speed and practical performance

Weaknesses and Challenges

• However, proximal point methods also have some limitations and challenges in practice, such as the difficulty of solving the subproblems exactly or efficiently
• The subproblems may be as hard as the original problem, especially when the proximal operator or the projection operator is not available in closed form or easy to compute (such as for matrix norms or graph-based regularizers)
• The choice of the regularization parameter can be critical for the performance of the algorithm, and there is no universal rule for selecting it optimally (it may require trial and error or adaptive strategies)
• The methods may suffer from slow convergence, especially when the problem is ill-conditioned (such as for inverse problems) or the operators are not strongly monotone or Lipschitz continuous
• The methods may be sensitive to errors or noise in the data or the computations, which can accumulate and propagate across the iterations and affect the quality of the solution (especially for inexact or stochastic variants of the algorithm)