# Solving this minimization problem

The goal is to approximate some function u known at some values s as a sum of function of the form $$( x - K)_{+}$$

Given $$(K_{i})_{1 \le i \le N_{k}} \in R^{+}$$ , $$(u_{i},s_{i})_{1 \le i \le N} \in R$$ ( can be restricted to some compact of the form $$]-S;S[$$ ) I want to solve the following problem $$\min_{\alpha \in R^{N_{k}}} || \sum_{i=1}^{N}(u_{i}-\sum_{j=1}^{N_{k}}\alpha_{j}(s_{i}-K_{j})_{+}||^{2}$$ under the N constraints $$u_{1} \le g(s_{1}) , u_{2} \le g(s_{2}) ,....,u_{n} \le g(s_{n})$$ where $$g(x)= \sum_{j=1}^{N_{k}}\alpha_{j}(x-K_{j})_{+}$$ ( I want my approximation to be at least always superior to my fonction u in each point )

Is there a solution to this problem ? Is it unique ? And if the solution can't be found analytically , what algorithm should i use?

• $u_i, s_i$ are given, i.e., inputs? What is $u(s_i)$? What is $g(s_i)$? Those are highly relevant to assessing the properties of this problem and how to solve it. If everything in the objective function (i.e., function being minimized) is given (i.e.. input), the objective function is linear least squares. As to what the constraints are? That depends on the answers to my questions. Aug 29, 2020 at 22:50
• I have edited , i hope its more clear , but I want my approximation in terms of sums to be always equal or greater than my fonction u in each point ( recall i dont know u , i only know some values at each point $s_{i}$) Aug 30, 2020 at 8:03