RandomPoint crashes kernel

Question

Bug introduced in 11.0 and fixed in 11.1

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Reported as CASE:3686963, 12 Aug 2016

I attempted to determine whether the problem with RandomPoint identified in my answer to question 89384 had been fixed in

$Version
(* 11.0.0 for Microsoft Windows (64-bit) (July 28, 2016) *)

In fact, a worse problem seems to have arisen. I began by attempting to reproduce the first plot in the question by

s = 0.36109;
t = 1*(1 - s);
equation1 = 1.505 < (1 - s)*8*(Sin[2*x]*Sin[z]*Cos[y] + Sin[2*y]*Sin[x]*Cos[z] +
    Sin[2*z]*Sin[y]*Cos[x]) - s*4*(Cos[2*x]*Cos[2*y] + Cos[2*y]*Cos[2*z] + 
    Cos[2*z]*Cos[2*x]) - t;
region1 = ImplicitRegion[equation1 && 6 <= x - y + 2*z <= 7, {{x, -5*Pi, 5*Pi}, 
    {y, -5*Pi, 5*Pi}, {z, -10, 10}}];
RandomPoint[region1, 10^4];

However, RandomPoint ran for about 30 minutes, consuming as much as 12 GB of memory, before the kernel crashed. This behavior is reproducible on my computer.

In contrast,

$Version
(* 10.4.1 for Microsoft Windows (64-bit) (April 11, 2016) *)

produces the result in the earlier question in a few seconds, as did version 10.2.0.

My questions are,

• Can others reproduce this behavior?
• Is it a bug?

Addendum

As István Zachar points out in a comment below, region1 can be displayed with RegionPlot3D, which takes a few minutes.

RegionPlot3D[region1, PlotPoints -> 200, BoxRatios -> {1, 1, 1}]

Although region1 is complicated, it does occupy a reasonable fraction of the plane on which it lies. Further, the plane itself can be represented in a few seconds by RandomPoint

region2 = ImplicitRegion[6 < x - y + 2*z < 7, 
    {{x, -5*Pi, 5*Pi}, {y, -5*Pi, 5*Pi}, {z, -10, 10}}];
pts2 = RandomPoint[region2, 10^4];
plt2 = ListPointPlot3D[pts2, Axes -> True, AxesLabel -> {"x", "y", "z"}, 
    BoxRatios -> {1, 1, 1}, 
    PlotRange -> {{x, -5*Pi, 5*Pi}, {y, -5*Pi, 5*Pi}, {z, -10, 10}}]

Yet, RandomPoint cannot find even a single point in region1.

pts1 = RandomPoint[region1]

Instead, it also fails in about 30 minutes after using over 12 GB of memory.

Answer 1

As part of its algorithm, RandomPoint attempts to determine the region dimension by calling

RegionDimension[region1]

In version 10.4, the above gives up quickly without returning a result, and RandomPoint proceeds by using a heuristic estimate. But in version 11.0, RegionDimension tries harder to compute the dimension, which is a rather costly symbolic computation going through heavy machinery like Reduce. It may be very slow and there is no guarantee of success, particularly in the case of a transcendental system.

Thank you for bringing up this example, the developers are aware of the issue and will look into possible improvements (for example, TimeConstrained could have been used to keep the runaway computation at bay).