12.3 Interior point methods for quadratic programming

Interior point methods revolutionize quadratic programming by traversing the feasible region's interior. Unlike the simplex method, they efficiently solve large-scale problems with polynomial-time complexity, making them ideal for portfolio and power system optimization.

The primal-dual approach simultaneously tackles primal and dual problems, leveraging KKT conditions. Newton's method and careful step size selection guide the algorithm along the central path, balancing optimality and feasibility to reach the optimal solution efficiently.

Interior Point Methods for Quadratic Programming

Concept of interior point methods

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• Interior point methods traverse feasible region's interior to find optimal solution
• Contrast with simplex method moving along boundary of feasible region
• Solve quadratic programming problems with quadratic objective function and linear constraints (portfolio optimization)
• Particularly effective for large-scale problems (power system optimization)
• Polynomial-time complexity ensures efficient solving of large problems
• Handle inequality constraints directly without conversion to equalities
• Central path guides algorithm towards optimal solution through feasible region's interior
• Barrier function penalizes approach to constraint boundaries maintaining feasibility (logarithmic barrier)

Primal-dual interior point method

• Simultaneously solves primal and dual problems exploiting complementarity conditions
• KKT conditions provide necessary and sufficient optimality criteria for convex QP
• Newton's method solves KKT system generating search directions for optimization
• Centering parameter balances optimality and feasibility during iterations
• Step size selection ensures iterates remain within feasible region interior
• Predictor-corrector variants like Mehrotra's algorithm improve convergence speed

Implementation of interior point algorithms

• Standard form: minimize $\frac{1}{2}x^TQx + c^Tx$ subject to $Ax = b, x \geq 0$
• Algorithm steps:

1. Initialize primal and dual variables
2. Compute residuals and duality gap
3. Form KKT system
4. Solve for search directions
5. Determine step size
6. Update variables
7. Check convergence criteria

• Efficient linear algebra routines crucial for performance (BLAS, LAPACK)
• Numerical considerations include handling ill-conditioning and scaling problem data

Convergence and complexity analysis

• Polynomial complexity: $O(\sqrt{n}L)$ iterations, $n$ variables, $L$ input size in bits
• Primal-dual gap typically decreases by constant factor each iteration
• Superlinear convergence near solution for faster final approach
• Computational complexity per iteration dominated by linear system solve $O(n^3)$ for dense problems
• Performance affected by problem size, sparsity structure, linear algebra accuracy, algorithm parameters

Interior point vs other techniques

• Simplex method better for small to medium-sized problems, exploits warm starts
• Active-set methods competitive for few active constraints, good for sequential quadratic programming
• Gradient projection methods effective for bound-constrained QP, simple implementation but slower convergence
• Augmented Lagrangian methods useful for nonlinear constraints, combinable with interior point methods
• Interior point methods excel in large-scale problems, dense constraint matrices, high-accuracy solutions
• Practical considerations include efficient implementations, robustness, warm-start capabilities