Interior point methods revolutionized linear programming by offering polynomial-time solutions. These algorithms follow a central path through the feasible region, balancing optimality and feasibility as they approach the solution.

Path-following algorithms, like short-step and long-step methods, navigate this central path. They use neighborhoods to stay close to the path, trading off step size and iteration count to efficiently reach the optimal solution.

Central Path and Neighborhood

Defining the Central Path

• Central path consists of optimal solutions to barrier problems for varying barrier parameters
• Connects analytic center of feasible region to optimal solution of linear program
• Characterized by primal-dual equations $Ax = b, A^Ty + s = c, XSe = \mu e$
• Barrier parameter μ controls distance from optimality
• As μ approaches zero, central path converges to optimal solution
• Visualized as smooth curve through interior of feasible region

Neighborhood and Its Importance

• Neighborhood of central path defines region around central path where iterates are considered "close"
• Typically defined using norm-based conditions ($\|XSe - \mu e\| \leq \gamma\mu$)
• Wide neighborhoods allow larger steps but may require more iterations
• Narrow neighborhoods restrict step size but often lead to faster convergence
• Trade-off between step size and number of iterations crucial for algorithm efficiency
• Neighborhoods ensure iterates maintain good properties (centrality, feasibility)

Step Methods

Short-Step Method Fundamentals

• Takes small steps along central path
• Guarantees polynomial-time convergence
• Uses narrow neighborhood (typically $\|XSe - \mu e\| \leq 0.1\mu$)
• Reduces duality gap by fixed factor in each iteration (typically 0.9)
• Simple to analyze theoretically
• May be slow in practice due to small step sizes

Long-Step Method Characteristics

• Allows larger steps compared to short-step method
• Uses wider neighborhood (typically $\|XSe - \mu e\| \leq 10\mu$)
• Can make more progress per iteration
• May require more iterations to converge in worst case
• Often performs better in practice despite weaker theoretical guarantees
• Balances larger steps with maintaining proximity to central path

Predictor-Corrector Method Implementation

• Combines advantages of short-step and long-step methods
• Consists of two phases per iteration: predictor step and corrector step
• Predictor step: Attempts large step towards optimality
• Corrector step: Brings iterate back closer to central path
• Often uses Mehrotra's predictor-corrector technique
• Adapts step sizes based on problem characteristics
• Generally outperforms both short-step and long-step methods in practice
• Requires more complex linear algebra computations per iteration

Interior Point Methods and Complexity

Polynomial-Time Complexity Analysis

• Interior point methods achieve polynomial-time complexity for linear programming
• Typically require $O(\sqrt{n}L)$ iterations, where n is problem dimension and L is input size
• Each iteration involves solving system of linear equations ($O(n^3)$ operations)
• Overall complexity: $O(n^{3.5}L)$ operations
• Compares favorably with simplex method's exponential worst-case complexity
• Practical performance often better than worst-case bounds suggest
• Complexity analysis uses potential functions to measure progress

Infeasible Interior Point Methods

• Start from infeasible points, gradually enforcing feasibility
• Useful when finding initial feasible point is difficult
• Modify standard primal-dual equations to handle infeasibility
• Introduce slack variables to relax constraints
• Minimize both original objective and infeasibility measure
• Converge to feasible solution if one exists, detect infeasibility otherwise
• Require careful handling of primal and dual infeasibility
• Often used in practice due to flexibility in starting points