Does anybody know which simplex-like algorithm CPLEX uses to solve quadratic programs. what is the so called Quadratic Simplex it is using?
Thank you in advance,
Mehdi
I'm not sure what CPLEX uses but the Simplex Method has been modified by Philip Wolfe to solve quadratic programming. In a nut shell, this is what it does:
Given a quadratic programming problem: QPP. p'x + 1/2x'Cx with constrains Ax = b
C has to be symmetric positive definite (positive semi-definite might work as well)
generate linear constraints using the Karush-Kuhn-Tucker conditions
modify the Simplex method in a way such that complementary slackness holds when choosing the pivot columns.
proceed with the other usual Simplex method steps
For more detailed information, please take a look at this paper:
http://pages.cs.wisc.edu/~brecht/cs838docs/wolfe-qp.pdf
Hope this helps.
Related
I was wondering if, given a min cost flow problem and an integer n, there is an efficient algorithm/package or mathematical method, to obtain the set of the
n-best basic solutions of the min cost flow problem (instead of just the best).
Not so easy. There were some special LP solvers that could do that (see: Ralph E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, 1986), but currently available LP solvers can't.
There is a way to encode a basis using binary variables:
b[i] = 1 if variable x[i] = basic
0 nonbasic
Using this, we can use "no good cuts" or "solution pool" technology to get the k best bases. See: https://yetanothermathprogrammingconsultant.blogspot.com/2016/01/finding-all-optimal-lp-solutions.html. Note that not all solution-pools can do the k-best. (Cplex can't, Gurobi can.) The "no-good" cuts work with any mip solver.
Update: a more recent reference is Craig A. Piercy, Ralph E. Steuer,
Reducing wall-clock time for the computation of all efficient extreme points in multiple objective linear programming, European Journal of Operational Research, 2019, https://doi.org/10.1016/j.ejor.2019.02.042
I'm trying to implement the Fast Messy GA using the paper by Goldberg, Deb, Kargupta Harik: fmGA - Rapid Accurate Optimization of Difficult Problems using Fast Messy Genetic Algorithms.
I'm stuck with the formula about the initial population size to account for the Building Block evaluation noise:
The sub-functions here are m=10 order-3(k=3) deceptive functions:
l=30, l'=27 and B is signal-to-noise ratio which is the ratio of the fitness deviation to the difference between the best and second best fitness value(30-28=2). Fitness deviation according to the table above is sqrt(155).
However in the paper they say using 10 order-3 subfunctions and using the equation must give you population size 3,331 but after substitution I can't reach it since I am not sure what is the value of c(alpha).
Any help will be appreciated. Thank you
I think I've figured it out what exactly is c(alpha). At least the graph drawing it against alpha looks exactly the same as in the paper. It seems by the square of the ordinate they mean the square of the Z-score found by Inverse Normal Random Distribution using alpha as the right-tail area. At first I was missleaded that after finding the Z-score it should be substituted in the Normal Random Distribution equation to fight the height(ordinate).
There is some implementation in Lua here https://github.com/xenomeno/GA-Messy for the interested folks. However the Fast Messy GA has some problems reproducing the figures from the original Goldberg's paper which I am not sure how to fix but these is another matter.
I am looking for the method or idea to solve the following optimization problem:
min f(x)
s.t. g(xi, yi) <= f(x), i=1,...,n
where x, y are variables in R^n. f(x) is convex function with respect to x. g(xi, yi) is a bunch of convex functions with respect to (xi, yi).
It is the problem of difference of convex functions (DC) optimization due to the DC structure of the constraints. Since I am fairly new to 'DC programming', I hope to know the global optimality condition of DC programs and the efficient and popular approaches for global optimization.
In my specific problem, it is already verified that the necessary optimality condition is g(xi*, yi*)=f(x*) for i=1,...,n.
Any ideas or solution would be appreciated, thanks.
For global methods, I would suggest looking into Branch and Bound, Branch and Cut, and Cutting Plane methods. These methods may be notoriously slow though depending on the problem size. It's because it is non-convex. It would be difficult to get efficient algorithms for global optimization for this problem.
For local methods, look into the convex-concave procedure. Actually, any heuristic might work.
I have a mixed integer quadratic program (MIQP) which I would like to solve using SCIP. The program is in the form such that on fixing the integer variables, the problem turns out to be a linear program. And on fixing the the continuous variables it becomes a Integer Program. A simple example :
max. \Sigma_{i} n_i * f_i(x_i)
such that.
n_1 * x_1 + n2 * x_2 < t
n_3 * x_1 + n2 * x_2 < m
.
.
many random quadratic constraints in n_i's and x_i's
so on
Here f_i is a concave piecewise linear function.
x_i's are continuous variables ( they take real values )
n_i's are integer variables
I am able to solve the problem using SCIP. But on problems with a large number of variables SCIP takes a lot of time to find the solution. I have particularly noticed that it does not find many primal solutions. Thus the rate at which the upper bound reduces is very slow. However, I could get better results by doing set heuristics emphasis aggressive.
It would be great if anyone can guide me on the following questions :
1) Is there any particular algorithm/ Software package which solves problems that fit perfectly into the model as described above ?
2) Suggestions on how to improve the rate at which primal solutions are found.
3) What type of branching can I use to get better results ?
4) Any guidance on improving performance would be really helpful.
I am okay with relaxing the integer constraints as well.
Thanks
1) The algorithm in SCIP should fit your problem. There are other software packages that implement similar algorithms, e.g., BARON and ANTIGONE.
2) Have a look which primal heuristics were successful in your run and change their parameters to run them more frequently.
3) No idea. Default should be ok.
4) Make sure that your variables have good bounds. Tighter bounds allow for a tighter relaxation to be constructed.
If you can post an instance of your problem somewhere, or a log of a SCIP run, including the detailed statistics at the end, maybe someone can give more hints on what to improve.
I am working with a convex QCQP as the following:
Min e'Ie
z'Iz=n
[some linear equalities and inequalities that contain variables w,z, and e]
w>=0, z in [0,1]^n
So the problem has only one quadratic constraint, except the objective, and some variables are nonnegative. The matrices of both quadratic forms are identity matrices, thus are positive definite.
I can move the quadratic constraint to the objective but it must have the negative sign so the problem will be nonconvex:
min e'Ie-z'Iz
The size of the problem can be up to 10000 linear constraints, with 100 nonnegative variables and almost the same number of other variables.
The problem can be rewritten as an MIQP as well, as z_i can be binary, and z'Iz=n can be removed.
So far, I have been working with CPLEX via AIMMS for MIQP and it is very slow for this problem. Using the QCQP version of the problem with CPLEX, MINOS, SNOPT and CONOPT is hopeless as they either cannot find a solution, or the solution is not even close to an approximation that I know a priori.
Now I have three questions:
Do you know any method/technique to get rid of the quadratic constraint as this form without going to MIQP?
Is there any "good" solver for this QCQP? by good, I mean a solver that efficiently finds the global optimum in a resonable time.
Do you think using SDP relaxation can be a solution to this problem? I have never solved an SDP problem in reallity, so I do not know how efficient SDP version can be. Any advice?
Thanks.