Convex Problem Examples

Optimization · Axiom Academy

EXAMPLE Convex Optimization Problems Worked examples demonstrating problem formulation, convexity verification, and solution approaches Convex optimization problems have a unique property: any local minimum is a global minimum. This guarantees that optimization algorithms will find the optimal solution efficiently. We'll explore three important applications from machine learning and finance. Problem Setup: An investor wants to allocate wealth among n assets to minimize portfolio variance while achieving a target expected return r_ . where ^n is the portfolio weight vector, ^ n n is the covariance matrix (positive semidefinite), and ^n is the vector of expected returns. The objective function is f( ) = 2 ^T . To check convexity, we compute the Hessian: Since is the covariance matrix of asset returns, it is positive semidefinite by construction. A function with positive semidefinite Hessian is convex. Conclusion: The objective function is convex. We need to check if the feasible region is a convex set: ^T = 1 is an affine set (convex) 0 defines the non-negative orthant (convex) The intersection of convex sets is convex, so the feasible region is convex. Conclusion: This is a convex optimization problem (convex objective, convex constraints). This is a Quadratic Programming (QP) problem with linear constraints. Interior-point methods: Efficient for medium to large-scale problems Active-set methods: Good for problems with few active constraints

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