For the variable $t$, problem is to find best multipliers $k$ which minimizes the objective function.
Time: $t_1$, $t_2$, $t_3$,... given in input
Multiplier $k_1$, $k_2$, $k_3$,... (These are continuous variables which needs to be found)
$c_1,c_2, \dots, c_N$ are constants
Main equation which has many cosine sine functions $k_1\sin(c_1x)+k_2\sin(c_2x)+k_3\sin(c_3x)+k_4\cos(c_1x)\dots$
Problem is to minimize sums of all equations below with best possible values of $(k_1, k_2, k_3,\dots)$. Also it is known that there is not an exact solution to the problem. So objective function is to minimize sum of errors: $\min(\text{Error}) = \min(\text{Error}(t_1) + \text{Error}(t_2) + \text{Error}(t_3)+\cdots)$,
when $x$ is $t_1$ --> $\text{Error}(t_1)=P_1-k_1\sin(c_1t_1)-k_2\sin(c_2t_1)-k_3\sin(c_3t_1)-k_4\cos(c_1t_1)\cdots$
when $x$ is $t_2$ --> $\text{Error}(t_2)=P_2-k_1\sin(c_1t_2)-k_2\sin(c_2t_2)-k_3\sin(c_3t_2)-k_4\cos(c_1t_2)\cdots$
when $x$ is $t_3$ --> $\text{Error}(t_3)=P_3-k_1\sin(c_1t_3)-k_2\sin(c_2t_3)-k_3\sin(c_3t_3)-k_4\cos(c_1t_3)\cdots$
P1 is a bound value of time variable. But $P(t)$ is not a analytic function.
I just have values for them, like when $t_1 = 5$, $P_1=0.7$, $t_2= 6$, $P_2= 0.3$ etc..
What is the best way to solve this problem? [considering i have thousands of lines of this equations.. Simply $(1000\times2000)$ matrices]. Which type of linear programming problem is this? Can linear algebra methods give a simple solution?