1. Minimizer 100067
2. Minimizer 100079

Method SLSQP uses Sequential Least SQuares Programming to minimize a function of several variables with any combination of bounds. 1.3, 0.7, 0.8, 1.9, 1.2. Minimize a multivariate function using Powell's Method. Minimizes a function of any number of variables using Powell's method with a full restart every n + 1 steps. It's useful and tested on a number of wikipedia's test functions for optimization, but it's not ready for scientific work.It uses a golden section line search so that it's maybe a bit slow but not.

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How do you minimize and maximize #f(x,y)=x/y-xy# constrained to #0<x-y<1#?

2 Answers

Explanation:

#lim_(x->oo)f(x,x-1/2)=lim_(x->oo)(x/(x-1/2)-x*(x-1/2))=-oo#
#lim_(y->0^+)f(y+1/2,y)=lim_(y->0^+)(y+1/2)/y-(y+1/2)y=+oo#

Both choices respect the constraint.

Explanation:

Battle epic games. Defining

#f(x,y)=x/y-x y#

and

#g_1(x,y,s)=x-y-s^2 = 0#
#g_2(x,y,s)=x-y-1+s^2=0#

the local maximization/minimization problem with inequality restrictions is transformed into an equivalent one, now with equality restrictions, so we can apply the Lagrange Multipliers technique for its resolution. The lagrangian is

#L(X,S,Lambda)=f(x,y)+lambda_1g_1(x,y,s_1)+lambda_2g_2(x,y,s_2)# https://yhsrqk.over-blog.com/2021/01/cain-and-abel-torrent.html.

Here #X = (x,y), S = (s_1,s_2), Lambda=(lambda_1,lambda_2)#

The stationary points are solutions of

#grad L =vec 0#

or

#{ (1/y - y + lambda_1 + lambda_2=0), ( x + x/y^2+ lambda_1 + lambda_2 = 0), (2 lambda_1 s_1 = 0), (2 lambda_2 s_2 = 0), ( x - y -s_1^2 = 0), ( x - y -1 + s_2^2= 0):}#

Solving for #X,S,Lambda# we obtain

Chatty for facebook messenger 2 1 download free. #(x = 0, y = -1, s_1 =pm1, s_2 = 0, lambda_1 = 0, lambda_2 = 0)#

Minimizer 100067

Here we observe that #s_2=0# so the active restriction is

#g_2(x,y,0)=0#

Minimizer 100079

Also we have

#(f @ g_2)(x)=1 + 1/(x-1) + x - x^2# and

#(d^2)/(dx^2)(f @ g_2)(x)=2/(x-1)^3-2# and
#(d^2)/(dx^2)(f @ g_2)(0)=-4#

This result shows that the point #x=0,y=-1# is a local máxima for the problem. Keyboard shortcut keys for macbook air.

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