Using Version 11's link to Ipopt, one can write a local minimization function which can take mulitple starting points and bounds on the variables and also return the Lagrange multipliers for the constraints and the variable bounds and the maximum constraint violation, as shown below
In[5]:= imin[
x^2 - y^2, {Cos[x - y] >= 0.5}, {{x, -5, {.2, 1},
5}, {y, -5, {-4, -1}, 5}}]
Out[5]= {{-24.9443, {x -> 0.235988, y -> -5.},
"conslagmult" -> {0 <= -0.5 +
Cos[x - y] <= \[Infinity] -> -0.54499},
"bndlagmult" -> {x -> {4.78912*10^-10, 5.25626*10^-10},
y -> {10.472, 2.50587*10^-10}}, "maxconsviol" -> 4.84126*10^-8,
"Solve_Succeeded"}, {-9.37535, {x -> -3.9528, y -> -5.},
"conslagmult" -> {0 <= -0.5 +
Cos[x - y] <= \[Infinity] -> -9.12861},
"bndlagmult" -> {x -> {2.39404*10^-9, 2.79887*10^-10},
y -> {2.0944, 2.50574*10^-10}}, "maxconsviol" -> 0,
"Solve_Succeeded"}}
Here are the corresponding results from FindMinimum
In[6]:= FindMinimum[{x^2 - y^2,
Cos[x - y] >= 0.5, -5 <= x <= 5, -5 <= y <= 5}, {{x, .2}, {y, -4}}]
Out[6]= {-24.9443, {x -> 0.235988, y -> -5.}}
In[7]:= FindMinimum[{x^2 - y^2,
Cos[x - y] >= 0.5, -5 <= x <= 5, -5 <= y <= 5}, {{x, 1}, {y, -1}}]
Out[7]= {-9.37535, {x -> -3.9528, y -> -5.}}
Attachments: