Constraint Optimization¶

Why It Matters¶

Theoretical optimization often produces impossible recommendations: short sell everything, use infinite leverage, hold negative positions. Real decisions always have constraints. This tool systematically incorporates constraints to find the best achievable outcome rather than the best theoretical outcome.

The Key Move¶

List all constraints explicitly: what are the binding limitations? (Can't invest more than 100%, need X minimum liquidity, regulatory caps, tax considerations, can't short, etc.) Formulate the objective function (maximize return for given risk). Use constrained optimization (quadratic programming for mean-variance, linear programming for simpler cases) to find allocations that maximize the objective while respecting all constraints. Check which constraints are binding—those are your bottlenecks.

Classic Application¶

Portfolio optimization with regulatory constraints. An insurance company must hold 30% government bonds (regulatory requirement), can't hold more than 5% in any single stock (concentration limit), and needs 10% in liquid assets (operational requirement). The optimal portfolio subject to these constraints differs significantly from the unconstrained theoretical optimum.

Surprising Application¶

Weekly schedule optimization. Objective: maximize productive output. Constraints: 8 hours sleep required, 2 hours family time (non-negotiable), 3 hours exercise/maintenance, 1-hour commute, meetings block 10 hours. Optimization reveals that your effective discretionary time is ~30 hours weekly. Allocate those optimally across priorities. The binding constraint (meetings) becomes obvious—if you could reduce meetings by 2 hours, that's where your biggest gain lies.

Failure Modes¶

Garbage in, garbage out—optimization requires accurate inputs for expected returns, risks, and correlations. Can find "solutions" that technically satisfy constraints but violate intent. Sensitive to input errors—small changes in expected returns can flip recommendations. Ignores robustness—optimal solution may be fragile. Assumes constraints are hard (but many real constraints have flexibility). Over-optimization leads to extreme positions.

Go Deeper¶

Grinold & Kahn, Active Portfolio Management, Chapter 14; Boyd & Vandenberghe, Convex Optimization, Chapter 4