### A Pluto.jl notebook ### # v0.19.46 using Markdown using InteractiveUtils # This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). macro bind(def, element) quote local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end local el = $(esc(element)) global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) el end end # ╔═╡ cdfd0cac-4c43-441d-bfb4-21ea5653a3a1 begin using Plots using Optim using OptimTestProblems using Roots using LineSearches using BenchmarkTools using PlutoUI end # ╔═╡ e6358a3b-f1d0-463c-a362-0451f0a20a8f html""" """ # ╔═╡ 110bde78-7381-4e0d-8ac8-4b878bd4de9b md""" # Optimization 2: Algorithms and some `Optim.jl` Applications Florian Oswald, 2024 """ # ╔═╡ 200201fd-1c4e-4242-9863-4970137d472a html"" # ╔═╡ e40bdc58-43a9-11ec-2c84-492ead8ce7c5 md""" ## Bracketing * A derivative-free method for *univariate* $f$ * works only on **unimodal** function $f$ * We can find a *bracket* in which the global minimum exists if we can find points $a < b < c$ such that $$f(a) \geq f(b) < f(c) \text{ or } f(a) > f(b) \leq f(c)$$ """ # ╔═╡ a691d7e0-84d4-4a69-ba41-68147254ff16 PlutoUI.LocalResource("./optimization2/unimodal.png") # ╔═╡ b7f8381c-3a9b-452e-9fc5-b7c9d4d69323 md""" ## Initial bracket * To start with, we need to identify first a region where bracketing will work. * Once we found a valid interval, we shrink it until we find the optimimum. * Here is an algorithm which will find a valid bracket, taking successive steps into a direction. * The function `bracket_minimum` (next slide) in action: """ # ╔═╡ d2a1181a-1146-478b-bcb2-cc763d6f211f PlutoUI.LocalResource("./optimization2/bracketing.png") # ╔═╡ 8db8d806-a6d1-4b98-9153-a92280de174a # algo 3.1 from Kochenderfer and Wheeler function bracket_minimum(f, x=0; s=1e-2, k=2.0) a, ya = x, f(x) b, yb = a + s, f(a + s) if yb > ya a, b = b, a ya, yb = yb, ya s = -s end while true c, yc = b + s, f(b + s) if yc > yb return a < c ? (a, c) : (c, a) end a, ya, b, yb = b, yb, c, yc s *= k end end # ╔═╡ 46533fde-5d5c-4a1a-a535-923672956f46 plot(sin,-5,10) # ╔═╡ 25c0a22e-af50-4455-93fd-d23f26b23d52 bracket_minimum(sin, -3) # ╔═╡ 08d053d9-c865-4ff3-a551-823b5fb41524 md""" ## Fibonacci Search * Suppose we have a bracket $[a,b]$ on $f$ and that we can only evaluate $f$ twice. * Where to evaluate it? * Let's split the interval into 3 equal pieces: (The next few pictures are ©️ Kochenderfer and Wheeler) """ # ╔═╡ df826745-a4c4-4301-af27-066fe8e634f5 PlutoUI.LocalResource("./optimization2/fibo1.png") # ╔═╡ 4cf52278-d871-4eb2-967d-d6368d01b0c8 md""" If we move the points to the center, we can do better: """ # ╔═╡ 62ed62ba-ce10-4d6d-b65f-0eb349465970 PlutoUI.LocalResource("./optimization2/fibo2.png") # ╔═╡ d2ce25bc-0aa8-419d-832e-4582ac83ba86 md""" * for $n$ queries of $f$, we get the following sequence * This can be determined analytically as $F_n = \frac{\psi^n - (1-\psi)^n}{\sqrt{5}}$ * and $\psi = (1+\sqrt{5})/2$ is called the *golden ratio*, which determines successive steps in the *Fibonacci Sequence*: $$\frac{F_n}{F_{n-1}} = \psi \frac{1-s^{n+1}}{1-s^n}, \quad s=\frac{1-\sqrt{5}}{1+\sqrt{5}}\approx-0.382$$ """ # ╔═╡ 39170a86-c216-447a-b6fc-367f0c1c82a8 PlutoUI.LocalResource("./optimization2/fibo3.png") # ╔═╡ 8e7d4e57-578a-4c25-8c37-65a3fcda7b40 md""" * *Golden Section Search* uses the *golden ratio* to approximate the Fibonacci sequence. * This trades off a minimal number of function evaluations with the best placement of evaluation points. """ # ╔═╡ dbac8440-659e-4a8d-a4b7-16bfd6d22c15 PlutoUI.LocalResource("./optimization2/golden1.png") # ╔═╡ d0b6dacf-8c13-4e31-a4a6-d4b5ee6a2f69 PlutoUI.LocalResource("./optimization2/golden2.png") # ╔═╡ adcd9256-650f-4d0e-8fe8-9b842b6b4c34 md""" ## `Optim.jl` Let's optimize a simple function with optim. """ # ╔═╡ a32820d0-5d91-4808-b942-202b760aec18 begin f(x) = exp(x) - x^4 plot(f,0,2, leg = :bottomleft, xlab = "x", ylab = "f(x)") end # ╔═╡ fd1fba34-1c4f-48f7-83b4-fd60c796ac17 md""" typically, we have a setup like ```julia function your_function(x) ... end using Optim optimize(your_function,from,to,which_method_to_use) ``` """ # ╔═╡ 1a3b8f05-aae9-4975-b8b5-9268145c1e82 begin minf(x) = -f(x) brent = optimize(minf,0,2,Brent()) golden = optimize(minf,0,2,GoldenSection()) vline!([Optim.minimizer(brent)],label = "brent") vline!([Optim.minimizer(golden)], label = "golden") end # ╔═╡ f127562c-d6c9-42c6-8cca-b1fd33ad4fbe md""" * Using Bracketing methods on non-unimodal functions is a *bad idea* * We identify a different optimum depending on where we start to search. """ # ╔═╡ dc4c2888-8ff1-417d-a3bd-c133fb3bbbff begin f2 = x->sin(x) + 0.1*abs(x) x_arr = collect(range(3π/2-8.5, stop=3π/2+5.5, length=151)) y_arr = f2.(x_arr) plot(x_arr,y_arr,legend = false) end # ╔═╡ 59b26c7c-f6c5-4ba6-a067-9d07f179fc12 md""" ub = $(@bind ub Slider(0:0.05:10, show_value = true, default = 3)) """ # ╔═╡ f615768b-f212-4f40-ac9c-9eb96c0d6ea1 function pbrent(ub) plot(x_arr,y_arr,legend = false) ob = optimize(f2, -2.5,ub,Brent()) scatter!([Optim.minimizer(ob)], [Optim.minimum(ob)], ms = 5, color = :red) end # ╔═╡ 1df4c4f2-6856-4431-afbb-df16541d22f7 pbrent(ub) # ╔═╡ fc015791-cd9e-4a1c-b6ea-4c70284bef64 md""" ## Bisection Methods * Root finding: `Roots.jl` finds points where a function equals 0. * Root finding in multivariate functions: [`IntervalRootFinding.jl`](https://github.com/JuliaIntervals/IntervalRootFinding.jl/) """ # ╔═╡ 32898baf-25d3-44cf-b92a-92e8ee320935 md""" up = $(@bind ubr Slider([3,4,5], show_value = true, default = 3)) """ # ╔═╡ 20664b8a-59d7-4d80-9926-7a982f72b276 let # if I write `let` it creates a "local scope", i.e. upon exiting at `end`, all objects are deleted f(x) = exp(x) - x^3 plot(f, -2,5,leg = false) hline!([0.0], ls = :dash, color=:black) xx = fzero(f,ubr-3,ubr) # from Roots.jl scatter!([xx],[f(xx)], color = :red, ms = 5) end # ╔═╡ 625e38ee-04c1-4792-bf3e-8689bdb4df3b md""" ## Running a Rosenbrock Banana through Optim * Let's get the rosenbrock banana and test it on several different algorithms! ### Comparison Methods * We will now look at a first class of algorithms, which are very simple, but sometimes a good starting point. * They just *compare* function values. * *Grid Search* : Compute the objective function at $G=\{x_1,\dots,x_N\}$ and pick the highest value of $f$. * This is very slow. * It requires large $N$. * But it's robust (will find global optimizer for large enough $N$) """ # ╔═╡ adb4ee25-44cb-4294-b410-f357e26336e1 collect(range(0, 3, length= 100)) # ╔═╡ 4a0dc0c1-ce50-425b-84ca-657c93b10966 begin prob = UnconstrainedProblems.examples["Rosenbrock"] ro = prob.f g! = prob.g! h! = prob.h! grid = collect(-1.0:0.01:3); # grid spacing is important! grid2D = [[i;j] for i in grid,j in grid]; val2D = map(ro,grid2D); r = findmin(val2D); md""" optimizer is at ($(grid2D[r[2]][1]), $(grid2D[r[2]][2])). """ end # ╔═╡ 7a8a9f51-49bc-4b6d-b68d-20f06dd7e350 md""" ## Local Descent Methods * Applicable to multivariate problems * We are searching for a *local model* that provides some guidance in a certain region of $f$ over **where to go to next**. * Gradient and Hessian are informative about this. ### Local Descent Outline All descent methods follow more or less this structure. At iteration $k$, 1. Check if candidate $\mathbf{x}^{(k)}$ satisfies stopping criterion: * if yes: stop * if no: continue 2. Get the local *descent direction* $\mathbf{d}^{(k)}$, using gradient, hessian, or both. 3. Set the *step size*, i.e. the length of the next step, $\alpha^k$ 4. Get the next candidate via $$\mathbf{x}^{(k+1)} \longleftarrow \alpha^k\mathbf{d}^{(k)}$$ ### The Line Search Strategy * An algorithm from the line search class chooses a direction $\mathbf{d}^{(k)} \in \mathbb{R}^n$ and searches along that direction starting from the current iterate $x_k \in \mathbb{R}^n$ for a new iterate $x_{k+1} \in \mathbb{R}^n$ with a lower function value. * After deciding on a direction $\mathbf{d}^{(k)}$, one needs to decide the *step length* $\alpha$ to travel by solving $$\min_{\alpha>0} f(x_k + \alpha \mathbf{d}^{(k)})$$ * In practice, solving this exactly is too costly, so algos usually generate a sequence of trial values $\alpha$ and pick the one with the lowest $f$. """ # ╔═╡ a6f1e909-9b87-42c7-b47a-d7190ba245bf PlutoUI.LocalResource("./optimization2/line-search.png") # ╔═╡ 2f970459-c1b1-418a-bab2-615aeba2e6c4 begin algo_hz = Newton(linesearch = HagerZhang()) # one particular linesearch optimize(ro, g!, h!, prob.initial_x, method=algo_hz) end # ╔═╡ 962e1fef-5ea1-4037-9e59-6b56c3f60909 md""" ### The Trust Region Strategy * First choose max step size, then the direction * Finds the next step $\mathbf{x}^{(k+1)}$ by minimizing a model of $\hat{f}$ over a *trust region*, centered on $\mathbf{x}^{(k)}$ * 2nd order Tayloer approx of $f$ is common. * Radius $\delta$ of trust region is changed based on how well $\hat{f}$ fits $f$ in trust region. * Get $\mathbf{x'}$ via $$\begin{align} \min_{\mathbf{x'}} &\quad \hat{f}(\mathbf{x'}) \\ \text{subject to } &\quad \Vert \mathbf{x}-\mathbf{x'} \leq \delta \Vert \end{align}$$ """ # ╔═╡ c031fe39-0f8c-4f6a-b851-266deb8f5b55 PlutoUI.LocalResource("./optimization2/trust-region.png") # ╔═╡ 3dab6ba2-7dd7-4f41-b1f3-e34d52211afb begin # Optim.jl has a TrustRegion for Newton (see below for Newton's Method) NewtonTrustRegion(; initial_delta = 1.0, # The starting trust region radius delta_hat = 100.0, # The largest allowable trust region radius eta = 0.1, #When rho is at least eta, accept the step. rho_lower = 0.25, # When rho is less than rho_lower, shrink the trust region. rho_upper = 0.75) # When rho is greater than rho_upper, grow the trust region (though no greater than delta_hat). res0 = Optim.optimize(ro, g!, h!, prob.initial_x, method=NewtonTrustRegion()) end # ╔═╡ 794092bf-cd6b-4cb8-aa11-c647157a0037 PlutoUI.LocalResource("./optimization2/trust-region2.png") # ╔═╡ 712a650d-4ada-4548-81ad-6c10b14d1a89 md""" ### Stopping criteria 1. maximum number of iterations reached 2. absolute improvement $|f(x) - f(x')| \leq \epsilon$ 3. relative improvement $|f(x) - f(x')| / |f(x)| \leq \epsilon$ 4. Gradient close to zero $|g(x)| \approx 0$ ## Gradient Descent The steepest descent is *opposite the gradient*. * Here we define $$\mathbf{g}^{(k)} = \nabla f(\mathbf{x}^{(k)})$$ * And our descent becomes $$\mathbf{d}^{(k)} = -\frac{\mathbf{g}^{(k)} }{\Vert\mathbf{g}^{(k)}\Vert }$$ * Minimizing wrt step size results in a jagged path (each direction is orthogonal to previous direction!) See Kochenderfer and Wheeler chapter 5, equation (5.6) $$\alpha^{(k)} = \arg \min{\alpha} f(\mathbf{x}^{(k)} + \alpha \mathbf{d}^{(k)})$$ """ # ╔═╡ 1117da28-0103-4264-95b2-07562c98103b res_gd = optimize(ro, g!,[-4,3.], GradientDescent(), Optim.Options(store_trace=true, extended_trace=true, iterations = 5000)) # ╔═╡ 45468faf-d2c1-4a68-997f-41f9879ede0f llevels() = exp.(range(log(0.5), stop = log(175.0), length = 20)) # ╔═╡ 11956a74-3ea2-491c-8eb1-d60db132c5c7 function plot_optim(res, start, x,y,ix; offset = 0) contour(x, y, (x,y)->sqrt(ro([x, y])), fill=false, color=:viridis, legend=false, levels = llevels()) xtracemat = hcat(Optim.x_trace(res)...) plot!(xtracemat[1, (offset+1):ix], xtracemat[2, (offset+1):ix], mc = :white, lab="") scatter!(xtracemat[1:1,ix], xtracemat[2:2,ix], mc=:black, msc=:red, lab="") scatter!([1.], [1.], mc=:red, msc=:red, lab="") scatter!([start[1]], [start[2]], mc=:yellow, msc=:black, label="start", legend=true) end # ╔═╡ cef266b6-7913-4abd-a665-03bd491049b0 md"5000 iteration = $(@bind i5k Slider(1:5000, show_value = true, default = 1))" # ╔═╡ 1a84b8d7-cfdc-420c-b4ff-d6cf32b82622 res_gd_zoom = optimize(ro, [-1,1.], GradientDescent(), Optim.Options(store_trace=true, extended_trace=true, iterations = 5000)); # ╔═╡ d0d68f47-95dd-4aba-8817-d015090f8c4c md"1000 iteration = $(@bind ik Slider(1:1000, show_value = true, default = 1))" # ╔═╡ 2cd50a99-f3d5-442d-9beb-cad56362ae1e md""" * pretty zig-zaggy * it may not be a huge issue for final convergence, but we take a lot of *unncessary* steps back and forth. * so in terms of computational cost, we incur quite a lot here. * zoom in even more: """ # ╔═╡ 28726ab6-747c-4537-94fd-df4f69ff0df5 md"zoom slider = $(@bind ikzoom Slider(120:160, show_value = true))" # ╔═╡ ca9fd4bb-f934-48df-8daf-f786013b6d69 let c = contour(-.85:0.01:-0.60, 0:0.01:1.2, (x,y)->sqrt(ro([x, y])), color=:viridis, legend=false, levels = exp.(range(log(1), stop = log(10.0), length = 70))) res = optimize(ro, [-4,-3.0], GradientDescent(), Optim.Options(store_trace=true, extended_trace=true)) xtracemat = hcat(Optim.x_trace(res)...) plot!(xtracemat[1, 120:ikzoom], xtracemat[2, 120:ikzoom], mc = :white, legend=true, label="Gradient Descent") scatter!(xtracemat[1:1,ikzoom], xtracemat[2:2,ikzoom], mc=:black, msc=:red, label = "") scatter!(xtracemat[1:1,120], xtracemat[2:2,120], mc=:blue, msc=:blue, label="start") end # ╔═╡ 7728ef9f-c84d-48b0-a25d-eb534f126d65 md""" ## Conjugate Gradient Descent * Tries to avoid the jagged path problem * It combines several methods: it minimizes a quadratic form of $f$ * The next descent direction uses the gradient *plus* some additional info from the previous step. """ # ╔═╡ 08339a83-73b6-4cb4-8b59-e30324fdfc8f md""" ## Second Order Methods ### Newton's Method * We start with a 2nd order Taylor approx over x at step $k$: $$q(x) = f(x^{(k)}) + (x - x^{(k)}) f'(x^{(k)}) + \frac{(x - x^{(k)})^2}{2}f''(x^{(k)})$$ * We form first order conditions (set it's root equal to zero) and rearrange to find the next step $k+1$: $$\begin{aligned} \frac{\partial q(x)}{\partial x} &= f'(x^{(k)}) + (x - x^{(k)}) f''(x^{(k)}) = 0\\ x^{(k+1)} &= x^{(k)} - \frac{f'(x^{(k)})}{f''(x^{(k)})} \end{aligned}$$ * The same argument works for multidimensional functions by using Hessian and Gradient * We would get a descent $\mathbf{d}^k$ by setting: $$\mathbf{d}^k = -\frac{\mathbf{g}^{k}}{\mathbf{H}^{k}}$$ * There are several options to avoid (often costly) computation of the Hessian $\mathbf{H}$: 1. Quasi-Newton updates $\mathbf{H}$ starting from identity matrix 2. Broyden-Fletcher-Goldfarb-Shanno (BFGS) does better with approx linesearch 3. L-BFGS is the limited memory version for large problems """ # ╔═╡ 5e8a328f-3624-4401-ad7a-f19bc94dd6cd function plot_optim(res1, res2, start, x,y,ix,lab1, lab2) contour(x, y, (x,y)->sqrt(ro([x, y])), fill=false, color=:viridis, legend=false, levels = llevels()) xtracemat = hcat(Optim.x_trace(res1)...) plot!(xtracemat[1, 1:ix], xtracemat[2, 1:ix], mc = :white, label=lab1, legend=true,lw=2) scatter!(xtracemat[1:1,ix], xtracemat[2:2,ix], mc=:black, msc=:red, label="") xtracemat2 = hcat(Optim.x_trace(res2)...) plot!(xtracemat2[1, 1:ix], xtracemat2[2, 1:ix], c=:blue, label=lab2,lw=2) scatter!(xtracemat2[1:1,ix], xtracemat2[2:2,ix], mc=:black, msc=:blue, label="") scatter!([1.], [1.], mc=:red, msc=:red, label="") scatter!([start[1]], [start[2]], mc=:yellow, msc=:black, label="start", legend=true) end # ╔═╡ 5a508717-8a42-466e-9415-ea6447290d19 plot_optim(res_gd, [-4,3.], -4.1:0.01:2, -4.5:0.01:4.1,i5k) # ╔═╡ e992331a-a739-45f7-9820-7ba6a6603cf9 plot_optim(res_gd_zoom, [-1,1], -2.5:0.01:2, -1.5:0.01:2,ik) # ╔═╡ 86c1c1df-9ee4-47a0-a40a-69fa3c245a94 begin res_newton = optimize(ro, g!, h!, [-1.,-1], Optim.Newton(), Optim.Options(store_trace=true, extended_trace=true)) res_gd_2 = optimize(ro, [-1.,-1], GradientDescent(), Optim.Options(store_trace=true, extended_trace=true, iterations = 10000)); end # ╔═╡ 37e8339e-cedc-46e4-8bcc-cae6bf2384a8 md"newton slider = $(@bind ikn Slider(1:20, show_value = true))" # ╔═╡ 6e297fb4-cf31-4928-9ec3-38b389fa6b58 plot_optim(res_gd_2, res_newton,[-1.,-1],-2.5:0.01:2, -1.5:0.01:2,ikn,"Gradient Descent", "Newton") # ╔═╡ 9ac7bf69-d1e2-46a4-96e8-4a8cb0ed3781 md""" ## Performance Let's look at some timings. """ # ╔═╡ c798c1a9-0f64-46fa-8844-3678c63708ef @benchmark optimize(ro, [0.0, 0.0], Optim.Newton(),Optim.Options(show_trace=false)) # ╔═╡ b62d94d4-4f64-4d3f-ae30-06e84013d3c7 @benchmark optimize(ro, g!, h!, [0.0, 0.0], Optim.Newton(), Optim.Options(show_trace=false)) # ╔═╡ 691f4695-c307-408f-892d-587b0b29a832 @benchmark optimize(ro, g!, h!, [-1.0, 3.0], BFGS()) # ╔═╡ e09184ba-da9d-43a2-bd6e-1a739e0722f5 @benchmark optimize(ro, g!, h!, [-1.0, 3.0], LBFGS()) # low memory # ╔═╡ 6d9670cf-5257-48af-8f62-3de6a88cbc51 md""" ## Direct Methods * No derivative information is used - *derivative free* * If it's very hard / impossible to provide gradient information, this is our only chance. * Direct methods use other criteria than the gradient to inform the next step (and ulimtately convergence). ### Cyclic Coordinate Descent -- Taxicab search * We do a line search over each dimension, one after the other * *taxicab* because the path looks like a NYC taxi changing direction at each block. * given $\mathbf{x}^{(1)}$, we proceed $$\begin{aligned} \mathbf{x}^{(2)} &= \arg \min_{x_1} f(x_1,x_2^{(1)},\dots,x_n^{(1)}) \\ \mathbf{x}^{(3)} &= \arg \min_{x_2} f(x_1^{(2)},x_2,x_3^{(2)}\dots,x_n^{(2)}) \\ \end{aligned}$$ * unfortunately this can easily get stuck because it can only move in 2 directions. ```julia ©️ Kochenderfer and Wheeler # start to setup a basis function, i.e. unit vectors to index each direction: basis(i, n) = [k == i ? 1.0 : 0.0 for k in 1 : n] function cyclic_coordinate_descent(f, x, ε) Δ, n = Inf, length(x) while abs(Δ) > ε x′ = copy(x) for i in 1 : n d = basis(i, n) x = line_search(f, x, d) end Δ = norm(x - x′) end return x end ``` """ # ╔═╡ 92212ec0-b822-4464-b346-dd221ad6f669 md""" ### General Pattern Search * We search according to an arbitrary *pattern* $\mathcal{P}$ of candidate points, anchored at current guess $\mathbf{x}$. * With step size $\alpha$ and set $\mathcal{D}$ of directions $$\mathcal{P} = {\mathbf{x} + \alpha \mathbf{d} \text{ for } \mathbf{d}\in\mathcal{D} }$$ * Convergence is guaranteed under the condition: $\mathcal{D}$ must be a positive spanning set: at least one $\mathbf{d}\in\mathcal{D}$ has a non-zero gradient. * We check for a distance $\alpha$ into the $\mathbf{d}$ direction for whether we can improve the objective. """ # ╔═╡ 4738c3b3-5c67-4ec6-8291-6426485a7832 PlutoUI.LocalResource("./optimization2/spanning-set.png") # ╔═╡ 27e77174-2232-4d67-ba8f-d9d944cce543 # ©️ Kochenderfer and Wheeler function generalized_pattern_search(f, x, α, D, ε, γ=0.5) y, n = f(x), length(x) evals = 0 while α > ε improved = false for (i,d) in enumerate(D) x′ = x + α*d y′ = f(x′) evals += 1 if y′ < y x, y, improved = x′, y′, true D = pushfirst!(deleteat!(D, i), d) break end end if !improved α *= γ end end # println("$evals evaluations") return (x,evals) end # ╔═╡ 496a4d17-0a31-4b6c-bae0-1461603b92e9 let D = [[1,0],[0,1],[-1,-0.5]] # 3 different directions generalized_pattern_search(ro,zeros(2),0.8,D,1e-6 ) end # ╔═╡ 5e289cc6-5528-42d9-a3cb-a0d9dac6a849 md""" ## Bracketing for Multidimensional Problems: Nelder-Mead * The Goal here is to find the simplex containing the local minimizer $x^*$ * In the case where $f$ is n-D, this simplex has $n+1$ vertices * In the case where $f$ is 2-D, this simplex has $2+1$ vertices, i.e. it's a triangle. * The method proceeds by evaluating the function at all $n+1$ vertices, and by replacing the worst function value with a new guess. * this can be achieved by a sequence of moves: * reflect * expand * contract * shrink movements. """ # ╔═╡ 4921b275-6773-4efa-91ce-02869c527dba PlutoUI.LocalResource("./optimization2/nelder-mead.png") # ╔═╡ b9d44daa-8781-4c55-b8ef-f0a8b1c92406 md""" * this is a very popular method. The matlab functions `fmincon` and `fminsearch` implements it. * When it works, it works quite fast. * No derivatives required. """ # ╔═╡ 9e0b8149-482d-4323-9670-a9756236fbeb @benchmark optimize(ro, [0.0, 0.0], NelderMead()) # ╔═╡ 51fa22fd-9d3b-40f1-a428-f0cdd8bb2374 md""" ## Stochastic Optimization Methods * Gradient based methods like steepest descent may be susceptible to getting stuck at local minima. * Randomly shocking the value of the descent direction may be a solution to this. * For example, one could modify our gradient descent from before to become $$\mathbf{x}^{(k+1)} \longleftarrow \mathbf{x}^{(k)} +\alpha^k\mathbf{g}^{(k)} + \mathbf{\varepsilon}^{(k)}$$ * where $\mathbf{\varepsilon}^{(k)} \sim N(0,\sigma_k^2)$, decreasing with $k$. * This *stochastic gradient descent* is often used when training neural networks. """ # ╔═╡ 2f6abe0f-43f5-40af-af51-561b76fcb29c md""" ### Simulated Annealing * We specify a *temperature* that controls the degree of randomness. * At first the temperature is high, letting the search jump around widely. This is to escape local minima. * The temperature is gradually decreased, reducing the step sizes. This is to find the local optimimum in the *best* region. * At every iteration $k$, we accept new point $\mathbf{x'}$ with $$\Pr(\text{accept }\mathbf{x'}) = \begin{cases} 1 & \text{if }\Delta y \leq0 \\ \min(e^{-\Delta y / t},1) & \text{if }\Delta y > 0 \end{cases}$$ * here $\Delta y = f(\mathbf{x'}) - f(\mathbf{x})$, and $t$ is the *temperature*. * We call $\Pr(\text{accept }\mathbf{x'})$ the **Metropolis Criterion**, building block of *Accept/Reject* algorithms. """ # ╔═╡ 7ff00636-cb5d-45a5-b3d3-011054902f8e # ©️ Kochenderfer and Wheeler # f: function # x: initial point # T: transition distribution # t: temp schedule, k_max: max iterations function simulated_annealing(f, x, T, t, k_max) y = f(x) ytrace = zeros(typeof(y),k_max) x_best, y_best = x, y for k in 1 : k_max x′ = x + rand(T) y′ = f(x′) Δy = y′ - y if Δy ≤ 0 || rand() < exp(-Δy/t(k)) x, y = x′, y′ end if y′ < y_best x_best, y_best = x′, y′ end ytrace[k] = y_best end return x_best,ytrace end # ╔═╡ 3cc0c7d6-0bd7-435c-abde-2c70f51d437e # enough banana. here is a function for big boys and girls to optimize! function ackley(x, a=20, b=0.2, c=2π) d = length(x) return -a*exp(-b*sqrt(sum(x.^2)/d)) - exp(sum(cos.(c*xi) for xi in x)/d) + a + exp(1) end # ╔═╡ e3ba1d6c-e781-4447-948c-51b4b6bc7813 packley() = surface(-30:0.1:30,-30:0.1:30,(x,y)->ackley([x, y]),cbar=false) # ╔═╡ fcbf2bf1-f037-4fb1-8f71-c512699173a9 packley() # ╔═╡ fb98ac86-bb03-49a5-a1a9-0eae38889823 anneal1 = Optim.optimize(ackley, [-30.0,-30], [30.0,150.0], [15.0,150], SAMIN(), Optim.Options(iterations=10^6)) # ╔═╡ f13056dc-3f17-438c-8b77-d8e1dc613c98 md""" * let's have Nelder Mead have a go at this as well. 😈 """ # ╔═╡ 83299874-e44b-4be3-8778-920df01581f4 nmackley = Optim.optimize(ackley, [-30.0,-30], [30.0,150.0], [15.0,150], NelderMead(), Optim.Options(iterations=10^6)) # ╔═╡ b43f3c20-aa3d-43ba-8a50-24645bd07ef1 let p0 = packley() scatter!([Optim.minimizer(anneal1)[1]],[Optim.minimizer(anneal1)[2]],[Optim.minimum(anneal1)],ms = 3, color = :red, label = "SAMIN", leg = :right) scatter!([Optim.minimizer(nmackley)[1]],[Optim.minimizer(nmackley)[2]],[Optim.minimum(nmackley)],ms = 3, color = :green, label = "Nelder-Mead") end # ╔═╡ ec549333-676d-4409-9256-8b4a8d8bf894 begin danger(head,text) = Markdown.MD(Markdown.Admonition("danger", head, [text])); danger(text) = Markdown.MD(Markdown.Admonition("danger", "Attention", [text])); info(text) = Markdown.MD(Markdown.Admonition("info", "Info", [text])); end # ╔═╡ 4717e637-07ab-4486-ac9a-c56b03ec7361 danger("Careful now.",md""" Lagarias et al. (SIOPT, 1999): At present there is no function in any dimension greater than one, for which the original Nelder-Mead algorithm has been proved to converge to a minimizer. Given all the known inefficiencies and failures of the Nelder-Mead algorithm [...], one might wonder why it is *used at all*, let alone *why it is so extraordinarily popular*. """) # ╔═╡ 8330b5bc-68fc-4d34-a8b3-1180e75f3d97 sb = md""" # """ # ╔═╡ dd1dc5d8-37c6-4263-9478-6112386d1b6e sb # ╔═╡ 28802a3b-2e75-4dee-ade6-c6361161c0e0 sb # ╔═╡ 8f45b52e-b7ce-4210-88fd-26ae5e09ae00 sb # ╔═╡ 43a0a115-5d53-48fc-af78-66f98e4f40ba sb # ╔═╡ 797442cc-3978-4749-907e-0627e81f93ad sb # ╔═╡ c026b58b-a46a-476f-aea3-74a5cd4aa867 sb # ╔═╡ ba6896c9-4e2e-4534-98c8-c7047db6d3fd sb # ╔═╡ a76474ab-056c-49f7-aff7-65d3e57417f6 sb # ╔═╡ c30d4d0e-9e00-41fe-8b92-f7b4d178146c sb # ╔═╡ f7fb3b3c-7c74-45eb-b1fb-c26170056d74 sb # ╔═╡ 29d84925-76ef-4238-8c17-fcc699daa80a sb # ╔═╡ f05bebbb-1635-46b2-8df2-bbefc71dcdf2 sb # ╔═╡ 220ba78a-eab9-4a98-9275-07046a7a267b sb # ╔═╡ 984a0439-fcf5-488b-8e4f-3b2ffac1bb2f sb # ╔═╡ 455f7670-6be7-47ba-9c31-7c3824e1bb68 sb # ╔═╡ 552d845e-817c-4e76-b699-37f149c6b5c1 sb # ╔═╡ 6f9b7197-cfc4-4f34-9be0-11d5578f56d7 sb # ╔═╡ 3b1c0e0a-a9b8-4ae8-9b31-a5d70e077201 sb # ╔═╡ 5e9ce511-f89b-437d-be55-5b0d390e3bae sb # ╔═╡ 7280b993-bfdd-48f4-9c1b-602e8dda8667 sb # ╔═╡ 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["Artifacts", "Libdl"] uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" version = "5.8.0+1" [[deps.libfdk_aac_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" version = "2.0.2+0" [[deps.libpng_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" version = "1.6.38+0" [[deps.libvorbis_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" version = "1.3.7+1" [[deps.nghttp2_jll]] deps = ["Artifacts", "Libdl"] uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" version = "1.52.0+1" [[deps.p7zip_jll]] deps = ["Artifacts", "Libdl"] uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" version = "17.4.0+2" [[deps.x264_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" version = "2021.5.5+0" [[deps.x265_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" version = "3.5.0+0" [[deps.xkbcommon_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" version = "1.4.1+0" """ # ╔═╡ Cell order: # ╟─e6358a3b-f1d0-463c-a362-0451f0a20a8f # ╟─110bde78-7381-4e0d-8ac8-4b878bd4de9b # ╟─200201fd-1c4e-4242-9863-4970137d472a # ╠═cdfd0cac-4c43-441d-bfb4-21ea5653a3a1 # ╟─e40bdc58-43a9-11ec-2c84-492ead8ce7c5 # ╟─a691d7e0-84d4-4a69-ba41-68147254ff16 # ╟─b7f8381c-3a9b-452e-9fc5-b7c9d4d69323 # ╟─d2a1181a-1146-478b-bcb2-cc763d6f211f # 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