Introduction

This notebook is parte of the talk on ParameterJuMP.jl in the third annual JuMP-dev workshop, held in Santiago, Chile, 2019

The main purpose of this notebook is to show an application of ParameterJuMP.jl. ParameterJuMP is well suited for Benders like decompositions therefore we shall try to demonstrate the usage of the library in one of the simplest problems that fits in the Benders decomposition framework. Norm-1 regression, which is a particular case of quantile regression is one of such problems.

Note that this is NOT the standard technique to solve Norm-1 regressions. Taylor made methods are available here for instance.

This notebook will require the following libraries:

ParameterJuMP itself

using ParameterJuMP

JuMP: the julia mathematical programming modeling tool

using JuMP

GLPK: A linear programing solver (other solvers could be used - such as Clp, Xpress, Gurobi, CPLEX and so on)

using Xpress
const OPTIMIZER = Xpress.Optimizer;

TimerOutputs: a time measuring library to demonstrate the advantage of using ParameterJuMP

using TimerOutputs

The following two julia default libraries

using LinearAlgebra # just use the dot function
using Random # to use random number generators

Plots library

using Plots
gr(); # plotting backend

Norm-1 regression

We will apply Norm-1 regression to the Linear Regression problem. Linear regression is a statistical tool to obtain the relation between one dependent variable and other explanatory variables. In other words, given a set of $n$ explanatory variables $X = \{ X_1, \dots, X_n \}$ we would like to obtain the best possible estimate for $Y$. In order to accomplish such a task we make the hypothesis that $Y$ is aapproximately linear function of $X$:

$Y = \sum_{j =1}^n \beta_j X_j + \varepsilon$

where $\varepsilon$ is some random error.

The estimation of the $\beta$ values relies on observations of the variables: $\{y^i, x_1^i, \dots, x_n^i\}_i$

In this notebook we will solve a problem where the explanatory variables are sinusoids of differents frequencies

First, we define the number of explanatory variables and observations

const N_Candidates = 700
const N_Observations = 5000
const N_Nodes = 1000

const Observations = 1:N_Observations
const Candidates = 1:N_Candidates
const Nodes = 1:N_Nodes
;

Initialize a random number generator to keep results deterministic

rng = Random.MersenneTwister(123);

Building regressors (explanatory) sinusoids

const X = zeros(N_Candidates, N_Observations)
const time = [obs / N_Observations * 1 for obs in Observations]
for obs in Observations, cand in Candidates
t = time[obs]
f = cand
X[cand, obs] = sin(2 * pi * f * t)
end

Define coefficients

β = zeros(N_Candidates)
for i  Candidates
if rand(rng)  (1-i/N_Candidates)^2 && i100
β[i] = 4*rand(rng)/i
end
end
println("First coefs: (β[1:min(10, N_Candidates)])") First coefs: [3.76206, 0.790906, 0.883406, 0.0521332, 0.0870966, 0.315345, 0.352853, 0.231924, 0.198475, 0.102393] Create noisy observations const y = X' * β .+ 0.1*randn(rng, N_Observations) plt = plot(time, y, xlabel = "Time (s)", ylabel = "Amplitude") plot!(plt, time, X'[:,1]) plot!(plt, time, X'[:,3]) plot!(plt, time, X'[:,9]) The classic tool to estimate linear regression models is the Least Squares method. The least squares method relies on solving the optimization problem: $\max \Bigg\{ \sum_{i \in Observations} \Big( y_i - \sum_{j \in Candidates} \beta_j x_{i,j} \Big) ^2 \Bigg\}$ In Norm-1 regression, the quadratic functions are replaced by absolute values: $\max\Bigg\{ \sum_{i \in Observations} \Big| y_i - \sum_{j \in Candidates} \beta_j x_{i,j} \Big| \Bigg\}$ This optimization problem can be recast as a Linear Programming Problem: \begin{align} & \min_{\varepsilon^{up}, \varepsilon^{dw}, \beta} && \sum_{i \in Observations} {\varepsilon^{up}}_i + {\varepsilon^{dw}}_i && \notag \\ & \text{subject to} && {\varepsilon^{up}}_i \geq + y_i - \sum_{j \in Candidates} \beta_j x_{i,j} && \forall i \in Observations \notag \\ & && {\varepsilon^{dw}}_i \geq - y_i + \sum_{j \in Candidates} \beta_j x_{i,j} && \forall i \in Observations \notag \\ & && {\varepsilon^{up}}_i, {\varepsilon^{dw}}_i \geq 0 && \forall i \in Observations \notag \\ \end{align} WhereObservations$is the set of all observations. This linear programming problem can be described in julia with JuMP # create an alias for the sum function (just for fun!) = sum # define the model function full_model_regression() time_build = @elapsed begin # measure time to create a model # initialize a optimization model full_model = Model(with_optimizer(OPTIMIZER)) # create optimization variables of the problem @variables(full_model, begin ɛ_up[Observations] 0 ɛ_dw[Observations] 0 β[1:N_Candidates] # 0 ≤ β[Candidates] ≤ 8 end) # define constraints of the model @constraints(full_model, begin ɛ_up_ctr[i in Observations], ɛ_up[i] + (X[j,i] * β[j] for j Candidates) - y[i] ɛ_dw_ctr[i in Observations], ɛ_dw[i] - (X[j,i] * β[j] for j Candidates) + y[i] end) # construct the objective function to be minimized @objective(full_model, Min, (ɛ_up[i] + ɛ_dw[i] for i Observations)) end # solve the problem time_solve = @elapsed optimize!(full_model) println("First coefficients in solution:$(value.(β)[1:min(10, N_Candidates)])")
println("Objective value: $(objective_value(full_model))") println("Time in solve:$time_solve")
println("Time in build: time_build") return nothing end full_model_regression (generic function with 1 method) Now we execute the functionthat builds the model and solves it N_Observations*N_Candidates < 10_000_000 && full_model_regression() First coefficients in solution: [3.76363, 0.796369, 0.878873, 0.0543967, 0. 0903491, 0.316708, 0.351747, 0.22979, 0.196402, 0.106295] Objective value: 362.6158214675506 Time in solve: 268.339451346 Time in build: 3.546058937 Benders decompositon Benders decompostions is used to solve large optimization problems with some special characteristics. LP's can be solved with classical linear optimization methods such as the Simplex method or Interior point methods provided by solvers like GLPK. However, these methods do not scale linearly with the problem size. In the Benders decomposition framework we break the problem in two pieces: A master and a slave problem. Of course some variables will belong to both problems, this is where the cleverness of Benders kicks in: The master problem is solved and passes the shared variables to the slave. The slave problem is solved with the shared variables FIXED to the values given by the master problem. The solution of the slave problem can be used to generate a constraint to the master problem to describe the linear approximation of the cost function of the shared variables. In many cases, like stochastic programming, the slave problems have a interesting structure and might be broken in smaller problem to be solved in parallel. We will descibe the decomposition similarly to what is done in: Introduction to Linear Optimization, Bertsimas & Tsitsiklis (Chapter 6.5): Where the problem in question has the form \begin{align} & \min_{x, y_k} && c^T x && + f_1^T y_1 && + \dots && + f_n^T y_n && \notag \\ & \text{subject to} && Ax && && && && = b \notag \\ & && B_1 x && + D_1 y_1 && && && = d_1 \notag \\ & && \dots && && \dots && && \notag \\ & && B_n x && && && + D_n y_n && = d_n \notag \\ & && x, && y_1, && && y_n && \geq 0 \notag \\ \end{align} Slave Given a solution for thexvariables we can define the slave problem as \begin{align} z_k(x) \ = \ & \min_{y_k} && f_k^T y_k && \notag \\ & \text{subject to} && D_k y_k && = d_k - B_k x \notag \\ & && y_k && \geq 0 \notag \\ \end{align} Thez_k(x)$function represents the cost of the subproblem given a solution for$x$. This function is a convex function because$xaffects only the right hand side of the problem (this is a standard resutls in LP theory). For the special case of the Norm-1 reggression the problem is written as: \begin{align} z_k(\beta) \ = \ & \min_{\varepsilon^{up}, \varepsilon^{dw}} && \sum_{i \in ObsSet(k)} {\varepsilon^{up}}_i + {\varepsilon^{dw}}_i && \notag \\ & \text{subject to} && {\varepsilon^{up}}_i \geq + y_i - \sum_{j \in Candidates} \beta_j x_{i,j} && \forall i \in ObsSet(k) \notag \\ & && {\varepsilon^{dw}}_i \geq - y_i + \sum_{j \in Candidates} \beta_j x_{i,j} && \forall i \in ObsSet(k) \notag \\ & && {\varepsilon^{up}}_i, {\varepsilon^{dw}}_i \geq 0 && \forall i \in ObsSet(k) \notag \\ \end{align} The collectionObsSet(k)$is a sub-set of the NObservations. Any partition of the NObservations collection is valid. In this notebook we will partition with the function: function ObsSet(K) obs_per_block = div(N_Observations, N_Nodes) return (1 + (K - 1) * obs_per_block):(K * obs_per_block) end ObsSet (generic function with 1 method) Which can be written in JuMP as follows. At this point we make a small detour to highlight the ParameterJuMP application. Every time you a find a IF block with the flag PARAM it means that we have two different implmentatins of the method: one relying on ParameterJuMP and the other using pure JuMP. function slave_model(PARAM, K) # initialize the JuMP model slave = if PARAM # special constructor exported by ParameterJuMP # to add the functionality to the model ModelWithParams(with_optimizer(OPTIMIZER)) else # regular JuMP constructor Model(with_optimizer(OPTIMIZER)) end # Define local optimization variables for norm-1 error @variables(slave, begin ɛ_up[ObsSet(K)] 0 ɛ_dw[ObsSet(K)] 0 end) # create the regression coefficient representation if PARAM # here is the main constructor of the Parameter JuMP packages # it will create model *parameters* instead of variables # variables are added to the optimization model, while parameters # are not. Parameters are merged with LP problem constants and do not # increase the model dimensions. β = Parameters(slave, zeros(N_Candidates)) else # Create fixed variables @variable(slave, β[1:N_Candidates] == 0) end # create local constraints # note that *parameter* algebra is implemented just like variables # algebra. We can multiply parameters by constants, add parameters, # sum parameters and varaibles and so on. @constraints(slave, begin ɛ_up_ctr[i in ObsSet(K)], ɛ_up[i] + (X[j,i] * β[j] for j Candidates) - y[i] ɛ_dw_ctr[i in ObsSet(K)], ɛ_dw[i] - (X[j,i] * β[j] for j Candidates) + y[i] end) # ATTENTION β[j] * X[j,i] Is much slower # create local objective function @objective(slave, Min, (ɛ_up[i] + ɛ_dw[i] for i ObsSet(K))) # return the correct group of parameters if PARAM return (slave, β) else return (slave, β, FixRef.(β)) end end slave_model (generic function with 1 method) Master Now that all pieces of the original problem can be representad by the convex$z_k(x)functions we can recast the problem in the the equivalent form: \begin{align} & \min_{x} && c^T x + z_1(x) + \dots + z_n(x) && \notag \\ & \text{subject to} && Ax = b && \notag \\ & && x \geq 0 && \notag \\ \end{align} However we cannot pass a problem in this for to a linear programming solver (it could be passed to other kinds of solvers). Another standart result of optimization theory is that a convex function an be represented by its supporting hyper-planes: \begin{align} z_k(x) \ = \ & \min_{z, x} && z && \notag \\ & \text{subject to} && z \geq \pi_k(\hat{x}) (x - \hat{x}) + z_k(\hat{x}), \ \forall \hat{x} \in dom(z_k) && \notag \\ \end{align} Then we can re-write (again) the master problem as \begin{align} & \min_{x, z_k} && c^T x + z_1 + \dots + z_n \notag \\ & \text{subject to} && z_i \geq \pi_i(\hat{x}) (x - \hat{x}) + z_i(\hat{x}), \ \forall \hat{x} \in dom(z_i), i \in \{1, \dots, n\} \notag \\ & && Ax = b \notag \\ & && x \geq 0 \notag \\ \end{align} Which is a linear program! However, it has infinitely many constraints !!! We can relax thhe infinite constraints and write: \begin{align} & \min_{x, z_k} && c^T x + z_1 + \dots + z_n \notag \\ & \text{subject to} && Ax = b \notag \\ & && x \geq 0 \notag \\ \end{align} But now its only an underestimated problem. In the case of our problem it can be written as: \begin{align} & \min_{\varepsilon, \beta} && \sum_{i \in Nodes} \varepsilon_i \notag \\ & \text{subject to} && \varepsilon_i \geq 0 \notag \\ \end{align} This model can be written in JUMP function master_model(PARAM) master = Model(with_optimizer(OPTIMIZER)) @variables(master, begin ɛ[Nodes] 0 β[1:N_Candidates] # 0 ≤ β[Candidates] ≤ 8 end) @objective(master, Min, (ɛ[i] for i Nodes)) sol = zeros(N_Candidates) return (master, ɛ, β, sol) end master_model (generic function with 1 method) The method to solve the master problem and query its solution is given here: function master_solve(PARAM, master_model) model = master_model β = master_model optimize!(model) return (value.(β), objective_value(model)) end master_solve (generic function with 1 method) Supporting Hyperplanes With these building blocks in hand, we can start building the algorithm. So far we know how to: • Solve the relaxed master problem • Obtain the solution for the\hat{x}$(or$\beta$in our case) Now we can: • Fix the values of$\hat{x}$in the slave problems • Solve the slave problem • query the solution of the slave problem to obtain the supporting hyperplane the value of$z_k(\hat{x})$, which is the objectie value of the slave problem and the derivative$\pi_k(\hat{x}) = \frac{d z_k(x)}{d x} \Big|_{x = \hat{x}}$the derivative is the dual variable associated to the variable$\hat{x}$, which results by applying the chain rule on the constraints duals. These new steps are executed by the function: function slave_solve(PARAM, model, master_solution) β0 = master_solution slave = model # The first step is to fix the values given by the master problem @timeit "fix" if PARAM # *parameters* can be set to new values and the optimization # model will be automatically updated β_p = model ParameterJuMP.setvalue!.(β_p, β0) else # JuMP also has the hability to fix variables to new values β_v = model β_v_ref = model fix.(β_v, β0) end # here the slave problem is solved @timeit "opt" optimize!(slave) # query dual variables, which are sensitivities # they represent the subgradient (almost a derivative) # of the objective function for infinitesimal variations # of the constants in the linear constraints @timeit "dual" if PARAM # we can query dual values of *parameters* π = dual.(β_p) else # or, in pure JuMP, we query the duals form # constraints that fix the values of our regression # coefficients π = dual.(β_v_ref) end # π2 = shadow_price.(β_fix) # @show ∑(π .- π2) obj = objective_value(slave) rhs = obj - dot(π, β0) return (rhs, π, obj) end slave_solve (generic function with 1 method) Now that we have cutting plane in hand we can add them to the master problem: function master_add_cut(PARAM, master_model, cut_info, node) master = master_model ɛ = master_model β = master_model rhs = cut_info π = cut_info @constraint(master, ɛ[node] (π[j] * β[j] for j Candidates) + rhs) end master_add_cut (generic function with 1 method) Algorithm wrap up The complete algorithm is • Solve the relaxed master problem • Obtain the solution for the$\hat{x}$(or$\beta$in our case) • Fix the values of$\hat{x}$in the slave problems • Solve the slave problem • query the solution of the slave problem to obtain the supporting hyperplane • add hyperplane to master problem • repeat Now we grab all the pieces that we built and we write the benders algorithm by calling the above function in a proper order. The macros @timeit are use to time each step of the algorithm. function decomposed_model(PARAM) reset_timer!() # reset timer fo comparision time_init = @elapsed @timeit "Init" begin println("Initialize decomposed model") # Create the mastter problem with no cuts println("Build master problem") @timeit "Master" master = master_model(PARAM) # initialize solution for the regression coefficients in zero println("Build initial solution") @timeit "Sol" solution = (zeros(N_Candidates), Inf) best_sol = deepcopy(solution) # Create the slave problems println("Build slave problems") @timeit "Slaves" slaves = [slave_model(PARAM, i) for i Candidates] # Save initial version of the slave problems and create # the first set of cuts println("Build initial cuts") @timeit "Cuts" cuts = [slave_solve(PARAM, slaves[i], solution) for i Candidates] end UB = +Inf LB = -Inf println("Initialize Iterative step") time_loop = @elapsed @timeit "Loop" for k in 1:80 # Add cuts generated from each slave problem to the master problem @timeit "add cuts" for i Candidates master_add_cut(PARAM, master, cuts[i], i) end # Solve the master problem with the new set of cuts # obtain new solution candidate for the regression coefficients @timeit "solve master" solution = master_solve(PARAM, master) # Pass the new candidate solution to each of the slave problems # Solve the slave problems and obtain cuttin planes # @show solution @timeit "solve nodes" for i Candidates cuts[i] = slave_solve(PARAM, slaves[i], solution) end LB = solution new_UB = (cuts[i] for i Candidates) if new_UB UB best_sol = deepcopy(solution) end UB = min(UB, new_UB) println("Iter =$k, LB = $LB, UB =$UB")

if abs(UB - LB)/(abs(UB)+abs(LB)) < 0.05
println("Converged!")
break
end
end
println("First coefficients in solution: $(solution[1:min(10, N_Candidates)])") println("Objective value:$(solution)")
println("Time in loop: $time_loop") println("Time in init:$time_init")

print_timer()

return best_sol
end
decomposed_model (generic function with 1 method)

Run benders decomposition with pure JuMP

GC.gc()
β1 = decomposed_model(false);
Initialize decomposed model
Build master problem
Build initial solution
Build slave problems
Build initial cuts
Initialize Iterative step
Iter = 1, LB = 0.0, UB = 356653.53178661753
Iter = 2, LB = 30.399676563064446, UB = 356653.53178661753
Iter = 3, LB = 86.26967456145769, UB = 356653.53178661753
Iter = 4, LB = 149.98183770027555, UB = 300.9817490670204
Iter = 5, LB = 204.95734750716665, UB = 267.5103302069098
Iter = 6, LB = 227.4023133882345, UB = 251.87819595553975
Iter = 7, LB = 233.85017559155568, UB = 241.6185092949669
Converged!
First coefficients in solution: [3.76401, 0.794833, 0.878023, 0.056713, 0.0
95081, 0.320758, 0.351928, 0.226681, 0.193147, 0.105357]
Objective value: 233.85017559155568
Time in loop: 149.186656266
Time in init: 17.09593682
─────────────────────────────────────────────────────────────────────────
Time                   Allocations
──────────────────────   ───────────────────────
Tot / % measured:          167s / 100%            8.79GiB / 100%

Section          ncalls     time   %tot     avg     alloc   %tot      avg
─────────────────────────────────────────────────────────────────────────
Loop                  1     149s  89.7%    149s   5.45GiB  62.1%  5.45GiB
solve master        7     120s  72.0%   17.1s   33.3MiB  0.37%  4.76MiB
solve nodes         7    27.4s  16.5%   3.92s   4.88GiB  55.6%   714MiB
fix           4.90k    11.3s  6.78%  2.30ms   3.61GiB  41.2%   773KiB
dual          4.90k    10.2s  6.14%  2.09ms   1.25GiB  14.3%   268KiB
opt           4.90k    5.80s  3.49%  1.18ms   13.5MiB  0.15%  2.81KiB
add cuts            7    1.97s  1.18%   281ms    542MiB  6.03%  77.4MiB
Init                  1    17.1s  10.3%   17.1s   3.33GiB  37.9%  3.33GiB
Cuts                1    13.1s  7.89%   13.1s   2.37GiB  27.0%  2.37GiB
opt             700    10.9s  6.56%  15.6ms   2.10GiB  24.0%  3.07MiB
dual            700    1.62s  0.97%  2.31ms    190MiB  2.11%   278KiB
fix             700    167ms  0.10%   239μs   39.7MiB  0.44%  58.1KiB
Slaves              1    3.85s  2.31%   3.85s   0.94GiB  10.8%  0.94GiB
Master              1   2.92ms  0.00%  2.92ms    871KiB  0.01%   871KiB
Sol                 1   6.42μs  0.00%  6.42μs   5.66KiB  0.00%  5.66KiB
─────────────────────────────────────────────────────────────────────────

Run benders decomposition with ParameterJuMP

GC.gc()
β2 = decomposed_model(true);
Initialize decomposed model
Build master problem
Build initial solution
Build slave problems
Build initial cuts
Initialize Iterative step
Iter = 1, LB = 0.0, UB = 356653.53178661753
Iter = 2, LB = 30.970893155886884, UB = 356653.53178661753
Iter = 3, LB = 85.03827213320974, UB = 356653.53178661753
Iter = 4, LB = 137.43316230115607, UB = 324.51973053567116
Iter = 5, LB = 198.2684305114623, UB = 264.5511437349908
Iter = 6, LB = 225.27179951526546, UB = 252.06650777298213
Iter = 7, LB = 233.04451775497623, UB = 244.32095435502725
Converged!
First coefficients in solution: [3.76189, 0.793195, 0.876522, 0.0542087, 0.
0937809, 0.31955, 0.351403, 0.226661, 0.194298, 0.105961]
Objective value: 233.04451775497623
Time in loop: 86.080175646
Time in init: 5.860248356
─────────────────────────────────────────────────────────────────────────
Time                   Allocations
──────────────────────   ───────────────────────
Tot / % measured:         91.9s / 100%            1.48GiB / 100%

Section          ncalls     time   %tot     avg     alloc   %tot      avg
─────────────────────────────────────────────────────────────────────────
Loop                  1    86.1s  93.6%   86.1s    661MiB  43.6%   661MiB
solve master        7    80.1s  87.2%   11.4s   33.3MiB  2.20%  4.76MiB
solve nodes         7    3.45s  3.76%   493ms   87.6MiB  5.78%  12.5MiB
opt           4.90k    3.00s  3.26%   612μs   54.3MiB  3.58%  11.4KiB
dual          4.90k    245ms  0.27%  49.9μs   30.4MiB  2.00%  6.34KiB
fix           4.90k    143ms  0.16%  29.2μs    383KiB  0.02%    80.0B
add cuts            7    2.48s  2.70%   355ms    539MiB  35.6%  77.1MiB
Init                  1    5.86s  6.37%   5.86s    855MiB  56.4%   855MiB
Slaves              1    3.42s  3.72%   3.42s    684MiB  45.2%   684MiB
Cuts                1    2.43s  2.65%   2.43s    170MiB  11.2%   170MiB
opt             700    1.85s  2.01%  2.64ms    106MiB  7.01%   155KiB
dual            700    327ms  0.36%   468μs   36.8MiB  2.43%  53.8KiB
fix             700   18.4ms  0.02%  26.3μs   54.7KiB  0.00%    80.0B
Master              1   8.03ms  0.01%  8.03ms    871KiB  0.06%   871KiB
Sol                 1   1.71μs  0.00%  1.71μs   5.66KiB  0.00%  5.66KiB
─────────────────────────────────────────────────────────────────────────

Plot resulting time series from the benders base estimations

const y1 = X' * β1
const y2 = X' * β2

plt = plot(time, y,
xlabel = "Time (s)", ylabel = "Amplitude")
plot!(plt, time, y1)
plot!(plt, time, y2) Acknowledgments

ParameterJuMP was developed by Joaquim Dias Garcia (@joaquimg) and Benoît Legat (@blegat)