Ipopt and Snopt in Strong Nonlinear System Long Trajectory Length Optimization Huawei Wang Human Motion & Control Lab Mechanical Engineering Cleveland State University Lab Meeting, July 23, 2018 Huawei Wang (HMC) Ipopt and Snopt in Long Period Strong Nonlinear System Optimization Lab Meeting 2018 0/8

Goal: Find out a suitable optimization platform and solver in strong nonlinear system long trajectory length optimization y Fy Huawei Wang (HMC) Ipopt and Snopt in Long Period Strong Nonlinear System Optimization Lab Meeting 2018 1/8

Optimization problem: y Fy Optimization problem: Mathematical model: y g external force (m, y, ydot) elastic force 𝑌 2 = 𝑌 1 𝑚 𝑌 2 =−𝑚𝑔+ 𝐹 𝑒 + 𝐹 𝑘 𝐹 𝑘 = −𝐾 𝑎 ∗𝑦, 𝑦>0 𝐹 𝑘 = − 𝐾 𝑔 𝑦 3 −𝐾 𝑎 ∗𝑦, 𝑦≤0 𝑌= 𝑦, 𝑦 𝑑𝑜𝑡 𝐾 𝑎 =0.1 𝑁 𝑚 𝐾 𝑔 =5𝑒7 𝑁 𝑚 3 which: 𝑚=1 𝑘𝑔 t (s) 1 2 3 4 5 …… (10, 0) (-0.02, ~) (8.2, 0) (9.4, 0) (8.3, 0) (9.7, 0) ground Optimization problem: Minimize external force 𝐹 𝑒 , while go through certain positions in a period of time. Huawei Wang (HMC) Ipopt and Snopt in Long Period Strong Nonlinear System Optimization Lab Meeting 2018 2/8

Optimization setting: y Boundary: y : ydot : Fe : [-0.1, 10] m [-20, 20] m/s [-100, 100] N g external force (m, y, ydot) elastic force t (s) 1 2 3 4 5 …… (10, 0) (-0.02, ~) (8.2, 0) (9.4, 0) (8.3, 0) (9.7, 0) Trajectory length: T : [20, 40, 60, …] s (10.3) Sampling rate: R : 100 Hz (5.0, 0) ground Huawei Wang (HMC) Ipopt and Snopt in Long Period Strong Nonlinear System Optimization Lab Meeting 2018 3/8

Optimization Methods: Ipopt Snopt Interior point optimizer Quadratic programming Linear solver: HSL solvers MUltifrontal Massively Parallel sparse direct Solver (Mumps) Parallel Sparse Direct Linear Solver (Pardiso) Interfaces: Snopta: friendly to new users Snoptb: more efficient, but nonlinear objectives and constraints need be define first. Snoptc: similar to Snoptb Huawei Wang (HMC) Ipopt and Snopt in Long Period Strong Nonlinear System Optimization Lab Meeting 2018 4/8

Optimization results: MA57, 20 s trajectory length Huawei Wang (HMC) Ipopt and Snopt in Long Period Strong Nonlinear System Optimization Lab Meeting 2018 5/8

Optimization results: Optimization time cost of each solver at different trajectory length Trajectory time 20 40 60 80 100 120 140 200 400 600 Problem Size var 6,000 12,000 18,000 24,000 30,000 36,000 42,000 60,000 120,000 180,000 NJ 14,000 28,000 56,000 70,000 84,000 98,000 140,000 280,000 420,000 Ipopt (MA57) 0.1h 0.48h 0.43h 0.925h 3.6h F1 -- Ipopt (MA86) 0.37h 0.55h 0.71h 1h 3.57h 8.7h Ipopt (Mumps) 0.47h 0.19h 0.58h 0.69h 3.42h Snopt F2 F1: maximum CUP time reached (16.7h) Time cost of MA86 of different trajectory length F2: maximum iteration reached (10*var) : CUP filled with tasks : only one task in CUP each time Huawei Wang (HMC) Ipopt and Snopt in Long Period Strong Nonlinear System Optimization Lab Meeting 2018 6/8

Optimization results of Ipopt F1: Ipopt MA57, 140 s trajectory length: Huawei Wang (HMC) Ipopt and Snopt in Long Period Strong Nonlinear System Optimization Lab Meeting 2018 7/8

Optimization results of Snopt F2: Snopt, 20 s trajectory length: Snopt, 2 s trajectory length: Huawei Wang (HMC) Ipopt and Snopt in Long Period Strong Nonlinear System Optimization Lab Meeting 2018 8/8