I have the following code, which is a bunch of linear constraints and tries to maximize a nonlinear objective using Maximize.

bounds=Table[b[i], {i, 128}];
boundCond=And@@Map[GreaterEqual[#,0]&, bounds];
upperCond=0.100897647440433502 + 0.0435742177069187164 b[2] + 0.110877290368080139 b[3] + 0.0627215802669525147 b[6] + 0.0303984060883522034 b[11] + 0.195413455367088318 b[12] + 0.00218466715887188911 b[13] + 0.178133592009544373 b[14] + 0.114736616611480713 b[16] + 0.0892148315906524658 b[19] + 0.0430798530578613281 b[20] + 0.127856075763702393 b[22] + 0.235492616891860962 b[24] + 0.100893214344978332 b[26] + 0.0550539642572402954 b[29] + 0.0691445842385292053 b[30] + 0.220859974622726440 b[33] + 0.0824572741985321045 b[34] + 0.111420944333076477 b[35] + 0.119025722146034241 b[36] + 0.0626275166869163513 b[37] + 0.0163705591112375259 b[38] + 0.163605749607086182 b[39] + 0.0159755386412143707 b[40] + 0.222821488976478577 b[41] + 0.0262936670333147049 b[42] + 0.176039412617683411 b[44] + 0.119316443800926209 b[46] + 0.00839051231741905212 b[48] + 0.0780717059969902039 b[49] + 0.0129324616864323616 b[51] + 0.160396441817283630 b[54] + 0.122262813150882721 b[55] + 0.225195974111557007 b[58] + 0.00371941062621772289 b[61] + 0.0782109126448631287 b[66] + 0.163961067795753479 b[67] + 0.0488019287586212158 b[68] + 0.0772711709141731262 b[70] + 0.0967618897557258606 b[72] + 0.0753138065338134766 b[74] + 0.138780370354652405 b[75] + 0.0779474973678588867 b[77] + 0.143815591931343079 b[78] + 0.216688260436058044 b[79] + 0.192488059401512146 b[82] + 0.0594606176018714905 b[83] + 0.0763093978166580200 b[84] + 0.197820603847503662 b[86] + 0.0981343537569046020 b[88] + 0.177558749914169312 b[89] + 0.0137362014502286911 b[91] + 0.0624699629843235016 b[93] + 0.172200247645378113 b[94] + 0.0640604123473167419 b[97] + 0.253324717283248901 b[99] + 0.0138462502509355545 b[102] + 0.00301329372450709343 b[103] + 0.102329082787036896 b[105] + 0.0432044677436351776 b[107] + 0.0967931896448135376 b[109] + 0.00924742128700017929 b[110] + 0.194597169756889343 b[112] + 0.0199854951351881027 b[115] + 0.00540251936763525009 b[117] + 0.0856299698352813721 b[118] + 0.00201270543038845062 b[119] + 0.0697343200445175171 b[120] + 0.0598938837647438049 b[121] + 0.115721419453620911 b[123] + 0.134053930640220642 b[126] + 0.0757555514574050903 b[128] <= 1.4
product:=Apply[Times, bounds];
Maximize[{product, upperCond && boundCond}, bounds];

Basically, bounds is a list of positive variables, and there is a linear constraint upperCond that involves some of the bounds variables, but not all. And I want to maximize the product of all bounds variables.

Now intuitively the maximal value of product should be unbounded because b[1], for example, could just be infinity as it does not appear in the linear constraint. But Mathematica gave me the following results:

Out[119]= {1.0149 10^-53, {b[1] -> 0.631844, b[2] -> 0.366275, b[3] -> 0.536577, b[4] -> 1.06186, b[5] -> 1.14916, b[6] -> 0.307205, b[7] -> 1.02957, b[8] -> 0.590336, b[9] -> 1.02625, b[10] -> 0.781978, 
b[11] -> 0.49093, b[12] -> 0.0258683, b[13] -> 0.585934, b[14] -> 0.0313637, b[15] -> 1.22059, b[16] -> 0.0865507, b[17] -> 0.744385, b[18] -> 0.652376, b[19] -> 0.191569, b[20] -> 0.401325, 
b[21] -> 0.591891, b[22] -> 0.0787587, b[23] -> 1.1079, b[24] -> 0.0211321, b[25] -> 0.673995, b[26] -> 0.270791, b[27] -> 1.09818, b[28] -> 1.15599, b[29] -> 0.847054, b[30] -> 0.286807, 
b[31] -> 1.12585, b[32] -> 1.12369, b[33] -> 0.0213811, b[34] -> 0.214411, b[35] -> 0.156409, b[36] -> 0.103377, b[37] -> 0.748765, b[38] -> 0.910576, b[39] -> 0.0460899, b[40] -> 0.943243, 
b[41] -> 0.0209113, b[42] -> 0.825903, b[43] -> 0.987693, b[44] -> 0.026583, b[45] -> 1.10995, b[46] -> 0.155892, b[47] -> 0.687417, b[48] -> 0.987659, b[49] -> 0.36043, b[50] -> 0.722281, 
b[51] -> 0.551261, b[52] -> 0.733565, b[53] -> 1.09543, b[54] -> 0.0416588, b[55] -> 0.596005, b[56] -> 0.599603, b[57] -> 1.16556, b[58] -> 0.0485591, b[59] -> 1.03535, b[60] -> 1.03462, 
b[61] -> 0.631659, b[62] -> 1.05244, b[63] -> 1.15654, b[64] -> 0.734218, b[65] -> 0.951136, b[66] -> 0.650616, b[67] -> 0.0792143, b[68] -> 0.354095, b[69] -> 0.559035, b[70] -> 0.295494, 
b[71] -> 0.627549, b[72] -> 0.153516, b[73] -> 0.578213, b[74] -> 0.375231, b[75] -> 0.0722742, b[76] -> 0.980545, b[77] -> 0.293393, b[78] -> 0.0657513, b[79] -> 0.028435, b[80] -> 1.06805, 
b[81] -> 1.03576, b[82] -> 0.0495011, b[83] -> 0.797685, b[84] -> 0.262131, b[85] -> 0.704682, b[86] -> 0.0488136, b[87] -> 0.730814, b[88] -> 0.129551, b[89] -> 0.0569268, b[90] -> 0.568371, 
b[91] -> 0.954501, b[92] -> 0.597521, b[93] -> 0.294341, b[94] -> 0.0337064, b[95] -> 1.1071, b[96] -> 0.97234, b[97] -> 0.254995, b[98] -> 0.71303, b[99] -> 0.0327845, b[100] -> 1.10424, 
b[101] -> 1.17713, b[102] -> 0.931491, b[103] -> 0.604491, b[104] -> 0.69352, b[105] -> 0.146641, b[106] -> 0.719575, b[107] -> 0.471206, b[108] -> 1.15883, b[109] -> 0.203697, b[110] -> 0.560459, 
b[111] -> 0.702077, b[112] -> 0.0634536, b[113] -> 1.11802, b[114] -> 1.08117, b[115] -> 0.583974, b[116] -> 0.987578, b[117] -> 0.655513, b[118] -> 0.214568, b[119] -> 0.652192, 
b[120] -> 0.426065, b[121] -> 0.307841, b[122] -> 0.996165, b[123] -> 0.130277, b[124] -> 1.02606, b[125] -> 1.17633, b[126] -> 0.0679813, b[127] -> 1.09422, b[128] -> 0.612329}}

Clearly Mathematica assigned a value to b[1], among other bounds variables that should have been unbounded, which leads to a "maximal" solution which should not exist.

But if I change the optimize objective from the product of all the bounds variables to the sum of all the bounds variables, it returned: NMaximize::ubnd: The problem is unbounded., which makes sense.

And also if I limit the number of bounds variables to say 5, it returned NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded., which again makes sense.

So seems like Maximize automatically uses NMaximize if constraints involve approximate real numbers, which clearly is the case here. So is the rounding error in NMaximize causing the problem when trying to find a numerical solution? And why is the rounding error not showing up in optimize for the sum of all the variables, and not showing up when the number of variables is small?