textbook sos example that worked on v0.7.1 and fails on v0.9.0

[RussTedrake]

We recently upgraded Drake from v0.7.1 to v0.9.0, and I experienced a regression in one of my "simple" examples using sums-of-squares for Lyapunov analysis. Mosek, SCS, and Clarabel v0.7.1 solved it fine.

For context, you can find the original notebook here. Unfortunately, the failure happens inside some alternations... specifically in the OptimizeLyapunov method at the bottom of the notebook, and if OptimizeMultipliers uses Mosek, and only OptimizeLyapunov uses Clarabel, then everything works.

Nevertheless, here is the Clarabel-only reproduction of the final solve which now results in Terminated with status = InsufficientProgress.

import clarabel
import numpy as np
from scipy import sparse

q = [1.8435820570082377, 1.8435820570082377, 2.7653730855123553, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
b = [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
P = sparse.csc_matrix((81, 81))

data = [-4.6713756474581505, -7.366046824401246e-16, -0.33482275518489235, 1.248525969168963e-19, 2.6328516974490616e-09, -3.9681215422316167e-16, -0.9587196352099683, -2.8174889909481824e-16, 2.6184498937672173e-07, -0.10625452869742728, 1.4101154172740503e-16, 0.5484683093767915, -1, 2.031287962092983e-21, -0.2742382564473529, -2.5439681835905437e-10, -1, -4.6713756474581505, -7.366046824401246e-16, -0.33482275518489235, 1.248525969168963e-19, 2.6328516974490616e-09, -4.6713756474581505, -1.1334168366632864e-15, -2.293542390394861, -2.8162404649790135e-16, 2.644778410741708e-07, 4.6713756474581505, 3.397925282169629e-16, 0.2698485912774968, -1.408622099643301e-16, 0.5484685685889291, 3.9681215422316167e-16, 2.523840753970691, 1.1593671545503101e-15, 0.6090525462693416, -1.248525969168963e-19, -2.6328516974490616e-09, 0.10625452869742728, 2.5580264378371873e-16, -0.8639869308685729, 2.8174889909481824e-16, -2.6184498937672173e-07, -2.031287962092983e-21, 0.38049278489038335, -1.4101154172740503e-16, -0.5484683093767915, 2.5439681835905437e-10, -2.031287962092983e-21, 0.2742382564473529, 2.5439681835905437e-10, -1.4142135623730951, -4.6713756474581505, -7.366046824401246e-16, -1.3348227551848924, 1.248525969168963e-19, 2.6328516974490616e-09, 4.6713756474581505, 3.397925282169629e-16, 0.37610311997492407, -2.8187375169173514e-16, 2.5921213767927265e-07, 3.9681215422316167e-16, 3.523840753970692, 1.159365123262348e-15, 0.8832908027166945, -1.248525969168963e-19, -2.6328516974490616e-09, 0.10625452869742728, 2.5580264378371873e-16, -0.8639869306141761, 2.8174889909481824e-16, -2.6184498937672173e-07, -2.031287962092983e-21, 0.38049278489038335, -1.4101154172740503e-16, -0.5484683093767915, 2.5439681835905437e-10, -2.031287962092983e-21, 0.2742382564473529, 2.5439681835905437e-10, -1, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 2, -1.4142135623730951, 1, -1, 2, -1.4142135623730951, 1, -1]
rows = [5, 6, 7, 8, 9, 12, 13, 14, 15, 19, 20, 21, 24, 25, 26, 30, 115, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 116, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 35, 117, 17, 37, 10, 38, 23, 40, 16, 43, 9, 47, 28, 52, 22, 58, 15, 65, 8, 73, 27, 82, 21, 92, 14, 103, 4, 39, 16, 41, 9, 44, 3, 48, 22, 53, 15, 59, 8, 66, 2, 74, 21, 83, 14, 93, 7, 104, 28, 42, 22, 45, 15, 49, 32, 54, 27, 60, 21, 67, 14, 75, 31, 84, 26, 94, 20, 105, 15, 46, 8, 50, 27, 55, 21, 61, 14, 68, 7, 76, 26, 85, 20, 95, 13, 106, 2, 51, 21, 56, 14, 62, 7, 69, 1, 77, 20, 86, 13, 96, 6, 107, 35, 57, 31, 63, 26, 70, 20, 78, 34, 87, 30, 97, 25, 108, 26, 64, 20, 71, 13, 79, 30, 88, 25, 98, 19, 109, 13, 72, 6, 80, 25, 89, 19, 99, 12, 110, 0, 81, 19, 90, 12, 100, 5, 111, 33, 91, 29, 101, 24, 112, 24, 102, 18, 113, 11, 114]
cols = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 43, 43, 44, 44, 45, 45, 46, 46, 47, 47, 48, 48, 49, 49, 50, 50, 51, 51, 52, 52, 53, 53, 54, 54, 55, 55, 56, 56, 57, 57, 58, 58, 59, 59, 60, 60, 61, 61, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 67, 67, 68, 68, 69, 69, 70, 70, 71, 71, 72, 72, 73, 73, 74, 74, 75, 75, 76, 76, 77, 77, 78, 78, 79, 79, 80, 80]
A = sparse.csc_matrix((data, (rows, cols)))
cones = [
  clarabel.ZeroConeT(37),
  clarabel.NonnegativeConeT(0),
  clarabel.PSDTriangleConeT(12),
  clarabel.PSDTriangleConeT(2),
]

settings = clarabel.DefaultSettings()
solver = clarabel.DefaultSolver(P, q, A, b, cones, settings)
solver.solve()

I know at least one relatively simple thing I can do to improve the numerics for this problem, but thought I should report the regression before I do.