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Minimize complex linear multi-parameter function in java

I need to minimize a complex linear multivariate function under certain restrictions .

Let x - array of complex numbers, length of L .

a [0], a [1], ..., a [L-1] are complex coefficients and

F is a complex function F (x) = x [0] * a [0] + x [1] * a [1] + ... + x [L -1] * a [L-1] , which should be minimized.

b [0], b [1], ..., b [L-1] are complex coefficients and there is a restriction

1 = complexConjuate (x [0]) * x [0] + complexConjuate (x [1]) * x [1] + ... + complexConjuate (x [L-1]) * x [L-1] , which must be performed.

I already looked at http://math.nist.gov/javanumerics/ in detail and looked at many documents. But I could not find a library that minimizes complex functions.

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You want to minimize a differentiable real-valued function fon a smooth hypersurface S. If such a minimum exists - in the situation after editing it is guaranteed to exist, because the hypersurface is compact - it occurs at the critical point of the restriction f|Sfrom fto S.

f, , M, - , gradient of f T(M) . .

, ( 1), () g(x) = 0 g, , f|S - x S, grad(f)|x grad(g)|x.

( ), ( ).

,

  • S, , (x|x) = 1, (a|b) a_1*b_1 + ... + a_k*b_k, g at x 2*x
  • L(x) = (c|x) = c_1*x_1 + ... + c_k*x_k, L x

, L ( c = 0, L ), , c , c/|c| -c/|c|.

, L(c/|c|) = 1/|c|*(c|c) = |c| L(-c/|c|) = -1/|c|*(c|c) = -|c|, -c/|c|, -|c|.

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