1.11 Constrained optimization
You can read a paper that uses Lagrangians without stopping.
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Constrained optimisation is maximising something subject to rules, and Lagrange multipliers are the standard machinery — needed to read the SVM dual form and a large share of ML papers without stalling. KKT conditions extend the idea to inequalities. It sits last in the calculus sequence as the advanced tool. The confusion worth fixing is treating the multiplier as an algebraic trick; it has a meaning — the sensitivity of the optimum to the constraint — and papers assume you know it.
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Equality constraints and Lagrange multipliers
Optimising subject to an equality restriction, using multipliers that convert the restricted problem into an unrestricted one. The multipliers turn out to have a meaning of their own, which is why the technique spreads.
Inequality constraints and KKT conditions
Extending the same idea to inequalities gives a set of conditions that any solution must satisfy. These conditions appear in the derivation of several classical methods.
Duality and why SVMs are stated in dual form
Every optimisation problem has a mirror-image problem, and sometimes the mirror is easier to solve. This is why one well-known classifier is nearly always presented in its mirrored form.
Projected gradient and penalty methods
Two practical ways to keep an iterative method inside its allowed region: push it back after each step, or add a cost for leaving. Both appear constantly in applied work.
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