How to solve such an optimization problem efficiently？?

Which algorithms are needed to solve an optimization problem?

• Please suggest name of some optimization algorithms.

• Answer:

  Depends on your optimization problem. Here are some algorithms you can google. 1. Simplex method for linear optimization. 2. Nelder-Mead simplex method for nonlinear optimization. 3. Steepest descent. Gradient descent. 4. Conjugate gradient method (for positive definite problems). 5. Bi-conjugate gradient method (for not positive definite problems). 6. Newton's method for optimization. Create a system of equations based on the  derivatives. Also see secant method. 7. Bracketing methods - including the bisection algorithm. 8. Quasi-Newton methods - see BFGS algorithm and limited memory BFGS.     also Davidon-Fletcher-Powell, DFP. 9. Newton and quasi-newton methods with a confidence region. 10. Particle swarm. 11. Genetic algorithms. 12. Simulated anealing. There are more, I am sure. Some methods work better  for different situations. A combinations of methods may be used also. Some methods require only function values and some require derivatives or approximate the derivatives. Basically optimization of a differentiable function is finding roots of the derivatives, so you will see a lot of root finding methods applied to optimization problems. Keep in mind also the difference in local and global optimums when you apply algorithms.