optimization-theory-expert skill

Overview

This skill is an optimization-theory expert that provides concise guidance on linear programming, convex optimization, gradient methods, and constrained optimization. It aggregates core techniques, common algorithmic choices, and practical diagnostics for modelers and researchers. Note: the skill is currently a legacy template and recommends validation against recent literature before production use.

How this skill works

The skill inspects problem structure (convexity, differentiability, constraint type) and suggests algorithms matched to that structure, such as simplex/Interior-Point for LP, proximal and accelerated methods for convex problems, and SQP or augmented Lagrangian for constrained nonlinear problems. It also outlines convergence diagnostics, step-size selection heuristics, and ways to test optimality and feasibility numerically. Users are prompted to verify recommendations with up-to-date sources when confidence is low.

When to use it

• Selecting a solver or algorithm for a new optimization model
• Diagnosing slow convergence or instability in iterative methods
• Designing preconditioning or scaling strategies for numerical stability
• Evaluating whether a problem is convex and identifying appropriate relaxations
• Preparing implementation choices for constrained nonlinear problems

Best practices

• Characterize the problem (convex vs nonconvex, smooth vs nonsmooth, constraints) before choosing methods
• Normalize and scale variables and constraints to improve numerical conditioning
• Use line-search or adaptive step-size rules rather than fixed steps for gradient methods
• Validate solutions with KKT conditions, dual residuals, and feasibility checks
• Document sources and update algorithm choices when new research affects guarantees

Example use cases

• Choosing between Interior-Point and simplex for large sparse linear programs
• Applying accelerated proximal gradient for L1-regularized convex objectives
• Designing an augmented Lagrangian workflow for nonlinear equality constraints
• Troubleshooting oscillation in gradient descent by inspecting Lipschitz estimates and step-size
• Formulating convex relaxations for combinatorial problems to obtain bounds

FAQ