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Please point me to relevant examples and/or APIs in ALGLIB.

I want to find the vector x whose objective function

• minimizes th L1 distance of x from a constant vector x0
• where L1 distance = sum of absolute values = sum abs((x(i)-x0(i))

and subject to this set of linear constraints:

• x(i+1) - x(i) >= constant(i) for i = 1..n-1 (i.e. x values in ascending order, with specified minimum spacing)
• x(1) >= min_constant
• x(n) <= max_constant

If I can get this working, then I want to try alternative distance metrics

• L2 distance = sum of squared differences = sum (x(i)-x0(i)**2
• L-infinity distance = max of absolute values = max abs((x(i)-x0(i))

Eventually, I want to extend the problem to two dimensions, with each x a point in the x-y plane.

I studied the manual and think I found the right examples:
For the L1 distance objective function which is discontinuous I will try:

Nonsmooth nonlinearly constrained optimization
https://www.alglib.net/translator/man/manual.cpython.html#example_minns_d_nlc

For the L2 distance objective function which is quadratic I will try:

Linearly constrained dense quadratic programming
https://www.alglib.net/translator/man/manual.cpython.html#example_minqp_d_lc1

I think L1 norm minimization can be cast as a linear programming problem.

Create non-negative dummy variables e_0l, e_0r, e_1l, e_1r, ...
then set equality constraints (in additional to the original constraints):
x(0) = x0(0) + e_0r
x(0) = x0(0) - e_0l
......

Then minimize e_0l + e0r + e_1l + e_1r + ...

So the LP example so be sufficient:
https://www.alglib.net/translator/man/manual.cpp.html#example_minlp_basic