Reference 0003

Portfolio Constraint Patterns

Common portfolio rules translated into solver-ready mathematical constraints.

Full investment

1' w = 1. All capital is allocated. This is an affine equality constraint.

Long-only

w >= 0. No short positions. This is an elementwise linear inequality.

Position bounds

l <= w <= u. Each asset has lower and upper weight limits.

Leverage cap

||w||_1 <= L. Gross exposure cannot exceed a limit. This is convex.

Turnover cap

||w - w0||_1 <= tau. The new portfolio cannot move too far from current holdings.

Factor exposure

l <= F' w <= u. Portfolio exposures to sectors, styles, or risk factors stay in allowed ranges.

Cardinality limit

number of nonzero weights <= k. This is usually nonconvex and often needs mixed-integer optimization or an approximation.

Feasible set

The set of all portfolios satisfying every constraint. Optimization only chooses from this set.

Constraint reading checklist

For each constraint, ask:
+ What real-world rule does it encode?
+ Is it equality or inequality?
+ Is it affine, convex, or nonconvex?
+ Could it make the problem infeasible?
+ Does it preserve the intended economic meaning?

Use this page while reading Lesson 0003.