HSG-MCS-HS21_Julia/JuliaTutorial-master/Tutorial_22a_Optimization_Optim.ipynb
2021-11-15 21:14:51 +01:00

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"# Numerical Optimization\n",
"\n",
"This notebook uses the [Optim](https://github.com/JuliaNLSolvers/Optim.jl) package which has routines for unconstrained optimization and for the case with simple bounds on the solution.\n",
"\n",
"As alternatives, consider the [NLopt](https://github.com/JuliaOpt/NLopt.jl) or the [JuMP](https://github.com/JuliaOpt/JuMP.jl) packages. They can easily handle various types of constrained optimization problems."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Load Packages and Extra Functions"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"printyellow (generic function with 1 method)"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Printf, Optim\n",
"\n",
"include(\"jlFiles/printmat.jl\")"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"using Plots\n",
"\n",
"#pyplot(size=(600,400))\n",
"gr(size=(480,320))\n",
"default(fmt = :svg)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Optimization with One Choice Variable\n",
"\n",
"Running \n",
"```\n",
"Sol = optimize(x->fn1(x,0.5),x₀,x₁)\n",
"```\n",
"finds the `x` value (in the interval `[x₀,x₁]`) that *minimizes* `fn1(x,0.5)`. The solution for the `fn1` function below should be $x=1.1$. The `x->fn1(x,0.5)` syntax makes this a function of `x` only.\n",
"\n",
"The output (`Sol`) contains a lot of information."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"fn1 (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function fn1(x,c) #notice: the function has two arguments\n",
" value = 2*(x - 1.1)^2 - c\n",
" return value\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
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],
"source": [
"x = -1:0.1:3\n",
"\n",
"p1 = plot( x,fn1.(x,0.5),\n",
" linecolor = :red,\n",
" linewidth = 2,\n",
" legend = nothing,\n",
" title = \"the fn1(x,0.5) function\",\n",
" xlabel = \"x\",\n",
" ylabel = \"y\" )\n",
"display(p1)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Results of Optimization Algorithm\n",
" * Algorithm: Brent's Method\n",
" * Search Interval: [-2.000000, 3.000000]\n",
" * Minimizer: 1.100000e+00\n",
" * Minimum: -5.000000e-01\n",
" * Iterations: 5\n",
" * Convergence: max(|x - x_upper|, |x - x_lower|) <= 2*(1.5e-08*|x|+2.2e-16): true\n",
" * Objective Function Calls: 6\n",
"\n",
"The minimum is at: 1.100\n",
"Compare with the plot above\n",
"\n"
]
}
],
"source": [
"Sol = optimize(x->fn1(x,0.5),-2.0,3.0)\n",
"println(Sol)\n",
"\n",
"printlnPs(\"\\nThe minimum is at: \", Optim.minimizer(Sol)) #the optimal x value\n",
"println(\"Compare with the plot above\\n\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## One Choice Variable: Supplying a Starting Guess Instead (extra)\n",
"\n",
"If you prefer to give a starting guess `x₀` instead of an interval, then supply it as as a vector `[x₀]`:\n",
"```\n",
"Sol = optimize(x->fn1(x[],0.5),[x₀],LBFGS()) \n",
"```\n",
"Notice: *(a)* `x[]` to make it a function of the first (and only) element in the vector `x`; *(b)* choose the `LBFGS()` method since the default method does not work in the case of only one choice variable."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The minimum is at: 1.100\n"
]
}
],
"source": [
"Solb = optimize(x->fn1(x[],0.5),[0.1],LBFGS())\n",
"\n",
"printlnPs(\"The minimum is at: \", Optim.minimizer(Solb))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Several Choice Variables: Unconstrained Optimization\n",
"\n",
"In the example below, we choose $(x,y)$ so as to minimize the fairly simple objective function\n",
"\n",
"$\n",
"(x-2)^2 + (4y+3)^2, \n",
"$\n",
"\n",
"without any constraints. The solution should be $(x,y)=(2,-3/4)$."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"fn2 (generic function with 1 method)"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function fn2(p)\n",
" (x,y) = (p[1],p[2]) #unpack the choice variables and get nicer names\n",
" L = (x-2)^2 + (4*y+3)^2\n",
" return L\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"nx = 2*41\n",
"ny = 2*61\n",
"x = range(1,stop=5,length=nx)\n",
"y = range(-1,stop=0,length=ny)\n",
"\n",
"loss2d = fill(NaN,(nx,ny)) #matrix with loss fn values\n",
"for i = 1:nx, j = 1:ny\n",
" loss2d[i,j] = fn2([x[i];y[j]])\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
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"source": [
"p1 = contour( x,y,loss2d', #notice: loss2d'\n",
" xlims = (1,5),\n",
" ylims = (-1,0),\n",
" legend = false,\n",
" levels = 21,\n",
" title = \"Contour plot of loss function\",\n",
" xlabel = \"x\",\n",
" ylabel = \"y\" )\n",
"scatter!([2],[-0.75],label=\"optimum\",legend=true)\n",
"display(p1)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"minimum at (x,y)= 2.000 -0.750\n"
]
}
],
"source": [
"Sol = optimize(fn2,[0.0;0.0]) #use p->lossfn(p,other arguments) if \n",
" #there are additional (non-choice) arguments\n",
"printlnPs(\"minimum at (x,y)= \",Optim.minimizer(Sol))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Several Choice Variables: Bounds on the Solutions\n",
"\n",
"The next few cells discuss how to impose bounds on the solution.\n",
"\n",
"In the example below, we impose $2.75 \\leq x$ (a lower bound) and $y \\leq -0.3$ (an upper bound). We will see that only one of these restrictions binds."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
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},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"p1 = contour( x,y,loss2d',\n",
" xlims = (1,5),\n",
" ylims = (-1,0),\n",
" legend = false,\n",
" levels = 21,\n",
" title = \"Contour plot of loss function\",\n",
" xlabel = \"x\",\n",
" ylabel = \"y\",\n",
" annotation = (3.5,-0.7,text(\"somewhere here\",8)) )\n",
"plot!([2.75,2.75],[-1,0.5],color=:red,linewidth=2,label=\"2.75 < x\",legend = true)\n",
"plot!([1,5],[-0.3,-0.3],color=:black,linewidth=2,label=\"y < -0.3\")\n",
"scatter!([2.75],[-0.75],color=:red,label=\"optimum\")\n",
"display(p1)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The optimum is at (x,y) = 2.750 -0.750\n"
]
}
],
"source": [
"lower = [2.75, -Inf]\n",
"upper = [Inf, -0.3]\n",
"\n",
"Sol = optimize(fn2,lower,upper,[3.0,-0.5])\n",
"printlnPs(\"The optimum is at (x,y) = \",Optim.minimizer(Sol))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Several Choice Variables: Supplying the Gradient (extra)\n",
"\n",
"Supplying a function for calculating the derivatives improves speed and accuracy. See below for an example. (The `inplace=false` means that the function for the derivatives creates a new matrix at each call.)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" * Status: success\n",
"\n",
" * Candidate solution\n",
" Final objective value: 7.296963e-30\n",
"\n",
" * Found with\n",
" Algorithm: L-BFGS\n",
"\n",
" * Convergence measures\n",
" |x - x'| = 9.34e-01 ≰ 0.0e+00\n",
" |x - x'|/|x'| = 4.67e-01 ≰ 0.0e+00\n",
" |f(x) - f(x')| = 8.75e-01 ≰ 0.0e+00\n",
" |f(x) - f(x')|/|f(x')| = 1.20e+29 ≰ 0.0e+00\n",
" |g(x)| = 2.13e-14 ≤ 1.0e-08\n",
"\n",
" * Work counters\n",
" Seconds run: 0 (vs limit Inf)\n",
" Iterations: 2\n",
" f(x) calls: 7\n",
" ∇f(x) calls: 7\n",
"\n"
]
}
],
"source": [
"function g2(x) #derivatives of fn2 wrt. x[1] and x[2]\n",
" G = [2*(x[1]-2), 2*4(4*x[2]+3)] #creates a new vector: use inplace = false in optimize()\n",
" return G\n",
"end\n",
"\n",
"Sol3 = optimize(fn2,g2,[1.0,-0.5],inplace=false)\n",
"println(Sol3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
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