Understanding Alternating Minimization in Convex Optimization: A Comprehensive Guide

Alternating minimization is a widely employed optimization technique in mathematical programming, particularly in convex optimization. This method is essential in various fields, including machine learning, statistics, and signal processing. The primary goal of this guide is to provide a detailed understanding of alternating minimization, including its key concepts, algorithm steps, convergence, applications, and advantages and considerations.

Problem Structure

Consider a convex function f(x, y) that depends on two sets of variables x and y. The objective is to minimize f(x, y) over both x and y.

Algorithm Steps

Initialization

Start with an initial guess for x and y denoted as x0 and y0.

Iterative Updates

The algorithm alternately optimizes one variable while keeping the other fixed. The process consists of the following steps:

tFix y: Minimize f(x, yk) with respect to x to obtain xk 1. tFix x: Minimize f(xk 1, y) with respect to y to obtain yk 1.

Repeat the above steps until convergence criteria are met, such as changes in x and y being below a certain threshold.

Convergence

Under specific conditions, such as convexity of the function and proper choice of initial points, alternating minimization can converge to a local minimum. For convex problems, it can converge to a global minimum. This makes it a powerful technique in optimization.

Applications

Alternating minimization is commonly used in various fields. Some notable applications include:

tMachine Learning: Matrix factorization problems tStatistics tSignal Processing

Example

Consider minimizing the function f(x, y) Ax - y^2 - λx^2 where A is a matrix, y is a vector, and λ is a regularization parameter. The algorithm would proceed as follows:

tFix y and minimize with respect to x. This gives the new value of x. tUpdate x and then fix x to minimize with respect to y. This gives the new value of y. tRepeat until convergence.

Advantages

The method allows for efficient handling of large-scale problems by breaking them into smaller, more manageable subproblems. Additionally, it is often simpler to implement compared to direct optimization of the joint variable set.

Considerations

Several factors can affect the performance of alternating minimization:

tThe choice of initial points can affect convergence. tThe method may converge to a local minimum rather than a global minimum in non-convex problems.

Conclusion

In summary, alternating minimization is a powerful technique in convex optimization that leverages the structure of the problem to iteratively improve variable estimates. This approach often leads to efficient solutions and is widely applicable in various fields.