I have 2 ordered sets $$X=\{X_1<\dots<X_n\}$$ and $$Y=\{Y_1<\dots<Y_m\}$$ with $X_1<Y_1$ and and $X_n<Y_m$. I wish to approximate an increasing continuous function $g$ by piecewise linear polynomials on knots $K_1<\dots<K_k$ such that $$\ell=-\sum_{i=1}^{n+m}\log(1+\lambda[g(Z_i)-1])+\sum_{i=1}^{n}\log(g(X_i))$$ where $\lambda$ is the solution of $$\sum_{i=1}^{n+m}\frac{g(Z_i)-1}{1+\lambda[g(Z_i)-1]}=0$$ and $$Z=X\cup Y.$$
I don't know yet either this problem has solution or not. I already tried using fmincon in matlab by initilizing $g$ as linear function. However, the result is far away from the true situation. Any advice?