KKT - KKT.nb 1 Part II Lagrange Multiplier Method...

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Part II: Lagrange Multiplier Method & Karush-Kuhn-Tucker (KKT) Conditions KKT Conditions General Non-Linear Constrained Minimum: Min: f[x] Constrained by: h[x] = 0 (m equality constraints) g[x] 0 (k inequality constraints) Introduce slack variables s i for the inequality contraints: g i [x] + s i 2 == 0 and construct the monster Lagrangian: L[x, l , m ] = f[x] + l h[x] + m i ( g i [x] + s i 2 ) Recall the geometry of the Lagrange multiplier conditions: The gradient of the objective function must be orthogonal to the tangent plane of the (active) constraints. That is the projection of the gradient of f onto the space of directions tangent to the constraint "surface" is zero. The KKT conditions are analogous conditions in the case of constraints. The KKT conditions are the following: 1) Gradient of the Lagrangian = 0 2) Constraints: h[x] = 0 (m equality constraints) & g[x] 0 (k inequality constraints) 3) Complementary Slackness ( for the s i variables) m .s == 0 4) Feasibility for the inequality constraints: s i 2 0 5) Sign condition on the inequality multipliers: m 0 One final requirement for KKT to work is that the gradient of f at a feasible point must be a linear combination of the gradients for the equality constraints and the gradients of the active constraints: this is often called regularity of a feasible point. At a feasible point for the constraints, the active constraints are those components of g with g i [x] = 0 ( if the value of the constraining function is < 0, that constraint is said to be inactive).
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