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Gradient descent To find the infimum of an arbitrary cost function, we use here the gradient descent. Let us consider the function $f(x,y)=2x^2+y^2-xy+2$ that we aim at minimizing on $ \mathbb{R}^2$.

#Install package import sys !{sys.executable} -m pip install numpy !{sys.executable} -m pip install matplotlib !{sys.executable} -m pip install amplpy from amplpy import AMPL # import pyAMPL from amplpy import ampl_notebook ampl = ampl_notebook( modules=["cplex"], # modules to install license_uuid="default", # license to use ) # instantiate AMPL object and register magics %%ampl_eval # define decision variables reset; # Declaration of optimization variables var xx; var yy; # Declaration of parameters param aa=-4; param bb=2; %%ampl_eval # Cost function minimize f: xx**2 + aa*(xx+yy) + 2*yy**2; # Constraints subject to g: xx+yy = bb; subject to h: xx >= 0; %%ampl_eval let xx:= 1; let yy:=2; # exhibit the model that has been built ampl.

Gradient descent with projection and augmented lagrangian algorithm To find the infimum of an arbitrary cost function, we use here the gradient descent. Let us consider the function $f(x,y)=0.5(x^2+(y-1)^2)$ that we aim at minimizing on $K\subset \mathbb{R}^2$ with $K={(x,y) | x= y}$.