Problem Solving Guide

Optimization · Axiom Academy

A practical framework for solving optimization problems from identification to verification. Linear Program (LP): Linear objective and constraints, continuous variables Quadratic Program (QP): Quadratic objective with linear constraints Integer Program (IP): Contains discrete decision variables (0/1, integer) Convex Problem: Convex objective over convex feasible set (local = global) Nonconvex Problem: Multiple local optima possible, harder to solve globally Decision Variables: What quantities are you choosing? Define domains clearly Objective Function: What are you maximizing or minimizing? Is it convex/concave? Constraints: What restrictions apply? Equality vs inequality? Linear vs nonlinear? Bounds: Are variables non-negative? Bounded above? Include box constraints Solution Verification Workflow Check Feasibility: Verify all constraints are satisfied at the solution point Test Optimality Conditions: For convex: check KKT conditions. For LP: check dual feasibility Evaluate Duality Gap: For convex problems, strong duality gives gap = 0 at optimum Perform Sensitivity Analysis: How does the solution change with constraint perturbations? Compare with Bounds: Check if solution lies within theoretical or computed bounds Numerical Instability: Poorly scaled problems cause solver failures. Always normalize data Local vs Global Optima: Nonconvex problems may converge to local optima. Try multiple starting points

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