KKT 与 SVMKKT & SVM

一个可手算的约束A small solvable constraint

最小化 1/2 w²，约束 w≥1，写作 g(w)=1−w≤0。最优点 w=1，乘子 λ=1；梯度条件 w−λ=0。它与线性可分硬间隔 SVM 的间隔约束有相同的基本结构。Minimize 1/2 w² subject to w≥1, written g(w)=1−w≤0. The optimum is w=1 with multiplier λ=1 and stationarity w−λ=0. This has the basic structure of a linearly separable hard-margin SVM constraint.

互补松弛Complementary slackness

λg(w)=0 表示约束严格不活跃时乘子为零；反过来约束位于边界并不保证乘子大于零。例如最小化 (w−1)² 且 w≥1，最优边界点可取 λ=0。λg(w)=0 means a strictly inactive constraint has zero multiplier. The reverse is not guaranteed: an active boundary can have zero multiplier. For example, minimize (w−1)² subject to w≥1; the optimum has λ=0.

必要与充分Necessity and sufficiency

可微凸问题中，满足 KKT 的可行点是全局最优；要保证最优点一定满足 KKT，还需要合适的约束资格。Slater 条件是凸问题常见的充分条件。For a differentiable convex problem, a feasible KKT point is globally optimal. Ensuring an optimum satisfies KKT also requires an appropriate constraint qualification; Slater’s condition is a common sufficient condition for convex problems.

公式与符号Formula and symbols

f 是目标，g 是不等式约束，w 是变量，λ 是乘子；四项依次是原始可行、对偶可行、驻点和互补松弛。f is the objective, g an inequality constraint, w the variable and λ the multiplier. The four conditions are primal feasibility, dual feasibility, stationarity and complementary slackness.

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支持向量机（SVM）Support vector machines (SVM)