A fully invested book must satisfy the budget , which an unconstrained optimum need not satisfy. Lagrange multipliers convert that equality into an extended stationary system; inequality constraints such as long-only weights add complementary slackness in the KKT conditions. A convex objective on a convex feasible set upgrades a local minimum to a global one. This note develops those tools with a portfolio budget as the running example.
Learning objectives. By the end of this note you should be able to:
- form a Lagrangian and solve a one-equality constrained problem;
- interpret as a shadow price;
- state the KKT idea for simple inequalities;
- recognise convex objectives / constraints so local minima are global, with a portfolio reading.
In this lesson
- Equality constraints and Lagrange
- Inequalities and the KKT idea
- Convexity: local becomes global
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
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