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Re: convex optimization problem
Posted:
Aug 27, 2012 8:33 AM


On 8/24/2012 5:14 PM, kumar vishwajeet wrote: > Alan_Weiss <aweiss@mathworks.com> wrote in message > <k17tbp$k7a$1@newscl01ah.mathworks.com>... >> On 8/24/2012 4:25 AM, kumar vishwajeet wrote: >> > Hi, >> > I have an objective function of the following type: >> > J = x log(x/k) + cx Subject to : f1(x) = alpha and f2(x) = beta >> > where alpha, beta, k and c are constants and x is a vector to be > >> solved for. So, it is a convex function. Which is the best > >> optimization routine in MATLAB to solve such problems. I'm currently >> > using fmincon and it is slower than snail. >> > >> > Thanks. >> This is a convex problem only with restrictions on the functions f1 >> and f2. I mean, there can be multiple local minima, depending on the >> functions f1 and f2. >> >> But as far as your question goes, the only solver in the toolbox to >> handle this type of problem is fmincon. If you are dissatisfied with >> its speed, you might want to try some of the suggestions in >> http://www.mathworks.com/help/toolbox/optim/ug/br44iv51.html#br44pif >> You might also want to try various algorithms, such as sqp. >> >> Good luck, >> >> Alan Weiss >> MATLAB mathematical toolbox documentation > > > I'm sorry. The actual cost function is : > J = summmation(i = 1:N) (x_i log(x_i/k_i) + c_i*x_i Subject to : x_i > >= 0 and summation(i=1:N) (x_i) = 1 > > where x_i and c_i are scalars. > Is it a good idea to change the constraint "x_i >= 0" to "x_i >= eps", > where eps is epsilon. I knew that x log x is not defined at x = 0. So > I intentionally made x log x = 0 for x = 0. This is valid for maximum > entropy problems. When using the sqp or interiorpoint algorithms there is no need to set x(i)>=eps; these algorithms obey bounds at all iterations, and your extension of the definition to the boundary is fine. http://www.mathworks.com/help/toolbox/optim/ug/brhkghv11.html#br9p_ry You might find that your problem runs faster if you provide gradients and, for the interior point algorithm, even a Hessian. If you have Symbolic Math Toolbox you can do this automatically: http://www.mathworks.com/help/toolbox/optim/ug/brn4nh7.html#brv_i_1 But for your problem, it is easy enough to calculate the gradients by hand.
Good luck,
Alan Weiss MATLAB mathematical toolbox documentation



