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mathematics_basic_for_ML
  • README
  • README
    • Summary
    • Geometry
      • EulerAngle
      • Gimbal lock
      • Quaternion
      • RiemannianManifolds
      • RotationMatrix
      • SphericalHarmonics
    • Information
      • Divergence
      • 信息熵 entropy
    • LinearAlgebra
      • 2D仿射变换(2D Affine Transformation)
      • 2DTransformation
      • 3D变换(3D Transformation)
      • ComplexTransformation
      • Conjugate
      • Hessian
      • IllConditioning
      • 逆变换(Inverse transform)
      • SVD
      • det
      • eigendecomposition
      • 矩阵
      • norm
      • orthogonal
      • special_matrix
      • trace
      • vector
    • Mathematics
      • Complex
      • ExponentialDecay
      • average
      • calculus
      • convex
      • derivative
      • 距离
      • function
      • space
      • Formula
        • euler
        • jensen
        • taylor
        • trigonometric
    • Numbers
      • 几何级数
      • SpecialNumbers
    • NumericalComputation
      • ConstrainedOptimization
      • GradientDescent
      • Newton
      • Nominal
      • ODE_SDE
      • Preprocessing
    • Probability
      • bayes
      • distribution
      • expectation_variance
      • 贝叶斯公式
      • functions
      • likelihood
      • mixture_distribution
      • 一些术语
      • probability_distribution
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convex

PreviouscalculusNextderivative

Last updated 2 years ago

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凸优化问题是指约束最优化问题:

minwf(w)s.t.gi(w)≤0,i=1,2,⋯ ,khi(w)=0,i=1,2,⋯ ,l\begin{aligned} min_w f(w) \\ s.t. g_i(w) \le 0, i = 1,2,\cdots,k \\ h_i(w) = 0, i = 1,2,\cdots,l \end{aligned}minw​f(w)s.t.gi​(w)≤0,i=1,2,⋯,khi​(w)=0,i=1,2,⋯,l​

其中, 目标函数f(w)和约束函数g(w)都是Rn上连续可微的凸函数。 约束函数h(w)是Rn上的仿射函数。

凸二次规划问题:当目标函数f(w)是二次函数且约束函数g(w)是仿射函数时,上述凸优化问题成为凸二次规划问题。

强凸问题:

f(y)≥f(x)+∇f(x)⊤(y−x)+μ2∣∣y−x∣∣f(y) \ge f(x) + \nabla f(x)^\top(y-x) + \frac{\mu}{2}||y-x||f(y)≥f(x)+∇f(x)⊤(y−x)+2μ​∣∣y−x∣∣