Lecture 12: convergence. Derivative (one variable)
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1 Lecture 12: convergence More about multivariable calculus Descent methods Backtracking line search More about convexity (first and second order) Newton step Example 1: linear programming (one var., one constr.) Example 2: linear programming (one var., two constr.) Example 3: linear programming (two var., one constr.) Example 4: linear programming (N var., M constr.) Derivative (one variable) f t 1
2 Derivative, i.e. gradient (multiple variables) General interpretation of derivative (gradient): First order approximation of the function (affine) Derivative f t 2
3 Second derivative General interpretation of second derivative: slope of the slope Hessian matrix (multiple variables) What if function has more than one variable? Example: 3
4 Descent methods, convex functions (reminder) [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ] Backtracking methods [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ] 4
5 Backtracking methods f Stopping criterion t Backtracking methods f Stopping criterion t 5
6 Backtracking methods f Stopping criterion t Backtracking methods f Stopping criterion t 6
7 Backtracking methods f Stopping criterion t Backtracking methods f Stopping criterion t 7
8 Backtracking methods f STOP t Backtracking methods [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ] 8
9 Convex functions: reminder [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ] First order conditions [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ] 9
10 Second order conditions [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ] Newton step Quadratic approximation of a function Graphical interpretation Numerical solution 10
11 Newton step Quadratic approximation of a function Graphical interpretation Newton step Quadratic approximation of a function Find the minimum of with respect to Newton step: 11
12 Newton step (more than one dimension) [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ] Newton step descent algorithm General algorithm: 12
13 Application: linear programming Back to linear programming: how would you solve a linear program with interior point methods? Instantiation of all the constraints: Application: linear programming Back to linear programming: how would you solve a linear program with interior point methods? Instantiation of all the constraints: 13
14 Linear programming (one variable, one constraint) Example: one constraint Rewrite the constraint Add logarithmic barrier Solve the unconstrained control problem: Linear programming (one variable, two constraints) Example: two constraints Rewrite the constraints Add logarithmic barrier Solve the unconstrained control problem: 14
15 Linear programming (two variables, two constraints) Example: two variables /two constraints Rewrite the constraints Add logarithmic barrier Solve the unconstrained control problem: Linear programming (two variables, one constraints) Example: two variables /two constraints Add logarithmic barrier Solve the unconstrained control problem: 15
16 Linear programming (N variables, M constraints) Example: two variables /two constraints Rewrite the constraints Linear programming (N variables, M constraints) Rewrite the constraints Add logarithmic barrier: 16
17 Linear programming (N variables, M constraints) Add logarithmic barrier: Gradient: Linear programming (N variables, M constraints) Gradient: Components of the gradient: 17
18 Linear programming (N variables, M constraints) Gradient: Components of the gradient: Linear programming (N variables, M constraints) Gradient: Components of the gradient: 18
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