How to solve optimization problem using sequential minimal optimization (SMO) in MATLAB

Hello all, I have the following optimization problem .

where is a column vector of dimension , is also a column vector of dimension and is matrix of dimension and is row vector of .

My query is How to solve it using sequential minimal optimization (SMO) in MATLAB ?

Any help in this regard will be highly appreciated.

Answer 1

[Matt J]

There is this offering on the File Exchange, which I have never used,

https://www.mathworks.com/matlabcentral/fileexchange/63100-smo-sequential-minimal-optimization?s_tid=srchtitle

It might be better just to use quadprog, although that uses a different algorithm.

[charu shree]

Thank you so much sir for your answer....

The first link seems to be not useful in my case...Will look for quadprog...

[charu shree]

I had checked the quadprog, but my query is how to solve it with those constraints....

[Torsten]

Here is the implementation of the constraints:

Aeq = b_l;
beq = zeros(size(b_l));
lb = zeros(size(b_l));
ub = C*ones(size(b_l));

[charu shree]

Thank you sir for your response...But how you have decided about the code which you have written.

Also what about summation.

[Matt J]

You could also use the problem-based set-up, e.g.,

K=rand(3); 
  K=K*K.';
b=rand(3,1)-0.5;
C=5;

alpha=optimvar('alpha',numel(b),'LowerBound',0,'UpperBound',C);

prob=optimproblem('Objective',alpha.'*(b.*K.*b')*alpha/2-sum(alpha),...
                  'Constraints', b'*alpha==0);

sol=solve(prob).alpha

Solving problem using quadprog.

Minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

sol = 3×1

5.0000 3.9981 5.0000

[charu shree]

Thank you sir for your detailed answer.

Basically, is the Gaussian radial basis function.

My query is that as both l, j in and is from 1 to , hence the norm inside exponential will always be zero, then how should we tackle this inside the two summations ?

[Matt J]

My query is that as both l, j in and is from 1 to L_t, hence the norm inside exponential will always be zero, then how should we tackle this inside the two summations ?

No, the norm will be non-zero whenever the vectors pr[l] and pr[j] are different. Also, this seems far afield now from your originally-posted question. Even if all the norms were zero, the ways to proceed with the optimization, which Torsten and I have already described, would not change.

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