#### Transcript Nonlinear Optimization - The Systems Realization Laboratory

What you can do for one variable, you can do for many (in principle) Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 78 Method of Steepest Descent The method of steepest descent (also known as the gradient method) is the simplest example of a gradient based method for minimizing a function of several variables. Its core is the following recursion formula: xk+1 = xk ±- k F k xk , xk+1 F(x) F Remember: Direction = dk = S(k) = -F(x(k)) = values of t he variables in the k and k+1 it eration. = object ive function to be minimized (or m axim ized) = gradients of the object ive function, const itut ing the direct ion of travel = t he size of t he step in the direct ion of travel Refer to Section 3.5 for Algorithm and Stopping Criteria Advantage: Simple Disadvantage: Seldom converges reliably. Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 79 Newton's Method (multi-variable case) How t o ext end Newt on’s method to multivariable case? xk +1 = xk - y’(xk) y’’(xk) Is t his correct ? No. Why? Start again wit h T aylor expansion: y(x) = y(xk) + y(xk)(x-xk) + 0.5 (x-xk)H(xk) (x-xk) T Remainder is dropped. Significance? See Sec. 1.4. Note t hatH is the Hessian containing t he second order derivatives. y(xk) xk +1 = xk - H(x ) k Is t his correct ? Not yet. Why? Newton’s m et hod for finding an extrem e point is xk +1 = xk - H-1(xk) y(xk) Don’t confuse H-1 with α. Optimization in Engineering Design Like the Steepest Descent Method, Newton’s searches in the negative gradient direction. Georgia Institute of Technology Systems Realization Laboratory 80 Properties of Newton's Method Good properties (fast convergence) if started near solution. However, needs modifications if started far away from solution. Also, (inverse) Hessian is expensive to calculate. To overcome this, several modifications are often made. • One of them is to add a search parameter in from of the Hessian. (similar to steepest descent). This is often referred to as the modified Newton's method. • Other modification focus on enhancing the properties of the second and first order gradient combination. • Quasi-Newton methods build up curvature information by observing the behavior of the objective functions and its first order gradient. This info is used to generate an approximation of the Hessian. Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 81 Conjugate Directions Method Conjugate direction methods can be regarded as somewhat in between steepest descent and Newton's method, having the positive features of both of them. Motivation: Desire to accelerate slow convergence of steepest descent, but avoid expensive evaluation, storage, and inversion of Hessian. Application: Conjugate direction methods are invariably invented and solved for the quadratic problem: Minimize: (½) xTQx - bTx Note: Condition for optimality is y = Qx - b = 0 or Qx = b (linear equation) Note: Textbook uses “A” instead of “Q”. Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 82 Basic Principle Definition: Given a symmetric matrix Q, two vectors d1 and d2 are said to be Q orthogonal or Q conjugate (with respect to Q) if d1TQd2 = 0. Note that orthogonal vectors (d1Td2 = 0)are a special case of conjugate vectors So, since the vectors di are independent, the solution to the nxn quadratic problem can be rewritten as x* = 0d0 + ... + n-1 dn-1 Multiplying by Q and by taking the scalar product with di, you can express in terms of d, Q, and either x* or b Note that A is used instead of Q in your textbook Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 83 Conjugate Gradient Method The conjugate gradient method is the conjugate direction method that is obtained by selecting the successive direction vectors as a conjugate version of the successive gradients obtained as the method progresses. Search direction @ iteration k. k 1 d k g k id i or d k 1 g k 1 k d k i 0 You generate the conjugate directions as you go along. Three advantages: 1) Gradient is always nonzero and linearly independent of all previous direction vectors. 2) Simple formula to determine the new direction. Only slightly more complicated than steepest descent. 3) Process makes good progress because it is based on gradients. Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 84 “Pure” Conjugate Gradient Method (Quadratic Case) 0 - Starting at any x0 define d0 = -g0 = b - Q x0 , where gk is the column vector of gradients of the objective function at point f(xk) 1 - Using dk , calculate the new point xk+1 = xk + k dk , where gkTd k k=- T d k Qd k Note that is calculated 2 - Calculate the new conjugate gradient direction dk+1 , according to: dk+1 = - gk+1 + k dk where gk+1 TQd k k= d kTQd k This is slightly different than your current textbook Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 85 Non-Quadratic Conjugate Gradient Methods • For non-quadratic cases, you have the problem that you do not know Q, and you would have to make an approximation. • One approach is to substitute Hessian H(xk) instead of Q. – Problem is that Hessian has to be evaluated at each point. • Other approaches avoid the Q completely by using Line Searches – Examples: Fletcher-Reeves and Polak-Robiere methods • Difference in methods: – find k through line search – different formulas for calculating k than the “pure” Conjugate Gradient algorithm Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 86 Polak-Robiere & Fletcher Reeves Method for Minimizing f(x) 0 - Starting at any x0 define d0 = -g0 ,where g is the column vector of gradients of the objective function at point f(x) 1 - Using dk , find the new point xk+1 = xk + k dk , where k is found using a line search that minimizes f(xk + k dk) 2 - Calculate the new conjugate gradient direction dk+1 , according to: dk+1 = - gk+1 + k dk where k can vary depending on what (update) formula you use. Fletcher-Reeves: Polak-Robiere: k ( g k 1 )T ( g k 1 ) T (g k ) (g k ) k ( g k 1 )T ( g k 1 g k ) ( g k )T ( g k ) Note: gk+1 is the gradient of the objective function at point xk+1 Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 87 Fletcher-Reeves Method for Minimizing f(x) 0 - Starting at any x0 define d0 = -g0 ,where g is the column vector of gradients of the objective function at point f(x) 1 - Using dk , find the new point xk+1 = xk + k dk , where k is found using a line search that minimizes f(xk + k dk) 2 - Calculate the new conjugate gradient direction dk+1 , according to: dk+1 = - gk+1 + k dk where T k ( g k 1 ) ( g k 1 ) ( g k )T ( g k ) See also Example 3.9 (page 73) in your textbook Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 88 Conjugate Gradient Method Advantages Attractive are the simple formulae for updating the direction vector. Method is slightly more complicated than steepest descent, but converges faster. See ‘em in action! For animations of each of ALL preceding search techniques, check out: http://www.esm.vt.edu/~zgurdal/COURSES/4084/4084-Docs/Animation.html Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 89