Which toolbox can solve my optimization problem

[Wanming Hao]

Hi, every one. 

   Now, I need to solve an optimzation problem as follows: In fact, My problem is to allocate power for maximizing sum rate 

Variables:  X=[x1,x2,...,xn], Y=[y1,y2,...,yn] and Z=[z1,z2,..,zn];  %%Power

maximize:  log(1+a1*x1)+log(1+a2*x2)+...+log(1+an*xn)+log(1+b1*y1)+log(1+b2*y2)+...+log(1+bn*yn)+log(1+c1*z1)+log(1+c2*z2)+...+log(1+cn*zn) %% sum rate

There are 4 constraint conditions: (A C B0 B1 Z0 Z1 are constants)

1. log(1+a1*x)>=A  x=x1, x2,..,xn;

2. sum(X)+sum(Y)+sum(Z)<=C;

3. B0<=log(1+b1*y1)+log(1+b2*y2)+...+log(1+bn*yn)<=B1

4. Z0<=log(1+c1*z1)+log(1+c2*z2)+...+log(1+cn*zn)<=Z1

In fact, the above optimization problem should be convex optimization problem, But when use the CVX toolbox, since the constraint 3, 4 include "Concave<=constant" form, this form is invalid constraint (Disciplined convex programming error:) in CVX toolbex.

Therefore, hope you can help me how to solve it. Thanks a lot in advance.

[Johan Löfberg]

log() <= is not a convex constraint

If you use YALMIP, any nonlinear solver will be applicable

[Wanming Hao]

Dear Johan

    Thank you very much for your help. I try to let YALMIP using a nonliner slover "'fmincon'" to solve my problem. But there is one error "Undefined function or method 'ge' for input arguments of type'logdet'" Namely, the constraint codition can not including "log" form, so if the problem can be solved by YALMIP? In addition, it seems impossible to transform the "log" form to other form.

Thanks a lot!

[Johan Löfberg]

logdet is a very specialized operator which only is used to define logdet objectives when you intend to solve the problem using a logdet aware SDP solver.

Your model above shows no use of log(det)) so I don't even see why you are talking about it

Of course you can have log in constraints

[Wanming Hao]

Dear Johan

Thank you very much for your help. Yes, my model only includes log, I will try it again. 

[Wanming Hao]

Dear Johan

    I try to write matlab code as follows. In my computer, it always make error, the reason is that "log" can not be used. Can you help me to analyze the reason?

Thanks a lot.

clear all

clc 

MacroUserSINR=[2 3 4 5 6];

MicroBsHeadSINR{1}=[4 5];

MicroBsHeadSINR{2}=[3 6];

MacroCellUserRateMin=1;

MicroCellUserNumber=4;

sdpvar MacroBsPowerVect(5,1) MicroBsHeadVect(2,2)

ObjMacro=0;

for i=1:5

   ObjMacro=ObjMacro+log(1+MacroBsPowerVect(i)*MacroUserSINR(i));

end

ObjMicro=0;

for m=1:2

    for i=1:2

        ObjMicro=ObjMicro+log(1+MicroBsHeadVect(m,i)*MicroBsHeadSINR{m}(i));

    end

end

Sum=ObjMacro+ObjMicro;   %%%%%%%%%%% Objective function

%%%%%%%%%%%%%%%%% the following are constraints

F=[];

for i=1:5

    F=[F,MacroBsPowerVect(i,1)>=(exp(MacroCellUserRateMin)-1)/MacroUserSINR(i)];  %%%constraints condition LP

end

F=[F,sum(MacroBsPowerVect)+sum(sum(MicroBsHeadVect))<=MacroBsPower];  %%%%% sum power constraint LP

for m=1:2

    MicroBsSumRate=0;

    for i=1:2

        MicroBsSumRate=MicroBsSumRate+log(1+MicroBsHeadVect(m,i)*MicroBsHeadSINR{m}(i));   %%%%%% constrainst condition NLP 

    end

    F=[F,MicroBsSumRate>=MicroCellUserRateMin*MicroCellUserNumber];   %%%%%%% namely A>=sum(log)>=B

    F=[F,MicroBsSumRate<=ClusterMaxRate(m)];

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

 ops=sdpsettings('solver','fmincon','verbose',0)

 sol = solvesdp(F,-Sum,ops)

[Johan Löfberg]

Undefined function or variable 'MacroBsPower'.

 

>> 

[Wanming Hao]

Sorry, I forgot it, it is 20; MacroBsPower=20

[Johan Löfberg]

Undefined function or variable 'MicroCellUserRateMin'.

 

>> 

[Wanming Hao]

Dear John

    Since I only cut one part of my code to run convenience for you. Maybe there are some undifiend paramneters, hope you can understand. MicroCellUserRateMin=1. Ther is another parameters ClusterMaxRate=[20,21]

Thank you very much again. 

Best wishes

[Johan Löfberg]

Runs without problems

>> ops=sdpsettings('solver','fmincon','verbose',1);

>> sol = solvesdp(F,-Sum,ops)

Your initial point x0 is not between bounds lb and ub; FMINCON

shifted x0 to strictly satisfy the bounds.

                                            First-order      Norm of

 Iter F-count            f(x)  Feasibility   optimality         step

    0       1   -9.380075e+00    4.000e+00    7.988e+00

    1       2   -1.514067e+01    1.765e+00    2.658e+00    1.365e+00

    2       3   -1.629282e+01    1.514e+00    2.117e+00    5.971e-01

    3       4   -1.924883e+01    9.127e-01    2.124e+00    2.492e+00

    4       5   -1.972554e+01    4.455e-01    1.399e+00    8.419e-01

    5       6   -2.020518e+01    0.000e+00    8.268e-01    8.131e-01

    6       7   -2.046445e+01    0.000e+00    2.734e-01    8.060e-01

    7       8   -2.057733e+01    0.000e+00    1.521e-01    9.151e-01

    8       9   -2.058359e+01    0.000e+00    9.992e-02    1.255e-01

    9      10   -2.063533e+01    0.000e+00    3.148e-02    1.156e-01

   10      11   -2.065082e+01    0.000e+00    1.985e-02    1.603e-01

   11      12   -2.065162e+01    0.000e+00    2.000e-02    5.867e-02

   12      13   -2.066613e+01    0.000e+00    5.021e-03    5.118e-02

   13      14   -2.066702e+01    0.000e+00    4.394e-03    1.771e-02

   14      15   -2.066716e+01    0.000e+00    4.000e-03    2.735e-02

   15      16   -2.067032e+01    0.000e+00    2.343e-03    1.828e-02

   16      17   -2.067036e+01    0.000e+00    1.759e-03    2.107e-03

   17      18   -2.067036e+01    0.000e+00    9.617e-04    4.281e-03

   18      19   -2.067036e+01    0.000e+00    8.000e-04    3.017e-03

   19      20   -2.067100e+01    0.000e+00    2.371e-04    4.489e-03

   20      21   -2.067101e+01    0.000e+00    1.600e-04    1.841e-03

   21      22   -2.067116e+01    0.000e+00    2.608e-05    8.202e-04

   22      23   -2.067116e+01    0.000e+00    1.930e-05    6.485e-05

   23      24   -2.067116e+01    0.000e+00    8.891e-06    1.191e-04

   24      25   -2.067116e+01    0.000e+00    4.227e-06    5.725e-05

   25      26   -2.067116e+01    0.000e+00    1.600e-06    3.664e-05

   26      27   -2.067117e+01    0.000e+00    3.200e-07    6.920e-06

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 

feasible directions, to within the selected value of the optimality tolerance,

and constraints are satisfied to within the selected value of the constraint tolerance.

<stopping criteria details>

sol = 

  struct with fields:

    yalmiptime: 0.4939

    solvertime: 0.3641

          info: 'Successfully solved (FMINCON)'

       problem: 0 

[Wanming Hao]

Dear John 

    Thank you very much. In my computer, it always appears the following error: Certainly, I have install the newest YALMIP  Matlab version is 7.8. 

??? Error: File: slog.m Line: 67 Column: 12

Expression or statement is incorrect--possibly

unbalanced (, {, or [.

Error in ==> sdpvar.log>check_for_special_cases at 82

    f = slog(x-1);

Error in ==> sdpvar.log at 11

        varargout{1} =

        check_for_special_cases(varargin{:});

Error in ==> Untitled at 12

   ObjMacro=ObjMacro+log(1+MacroBsPowerVect(i)*MacroUserSINR(i)); 

[Johan Löfberg]

Are you using the most recent version of YALMIP, and are you using a modern version of MATLAB?

[Wanming Hao]

I use the newest version of YALMIP and my version of MATLAB is R2009a (7.8.0)

[Wanming Hao]

anyway, "sdpt3" can be used in my computer, I use it for other problem.

[Johan Löfberg]

I guess that old version doesn't support unused output arguments. Hence you have to fix slog.m and explicitly use output names

[unused,common] =

there might be several places where yalmip crashes for the same reason

[Wanming Hao]

I do it according to your advice, modified as "function [unused,common]= slog(varargin)"   But, it still does not work. and the problem is following;

 for i = 1:length(g)

         gi = extsubsref(g,i);       

         [~,common] = find(mt(getvariables(gi),:) < 0);  %%%%% It reminders that  this line has problem

         y = recover(common(1));

         x = gi*y;

         if length(g)==1

             % Testing some display issues

             f =slogfrac([x;y]);

         else

            f = [f;slogfrac([x;y])];

         end

[Johan Löfberg]

why are you talking about sdpt3?

[Johan Löfberg]

?? 

I just told you have to fix that line

[Wanming Hao]

Dear John

    Thank you very much for your help. I talked about "sdpt3" in order to say the "YALMIP" can work.  Anywhere, it can work, but the solve can not be found: It reminders follows:   I use the same test code I sent you.

sol = 

    yalmiptime: NaN

    solvertime: NaN

          info: [1x196 char]

       problem: 9

[Johan Löfberg]

Turn on the debug option

[Wanming Hao]

Yes, the results is:

ops = 

                   solver: 'fmincon'

                  verbose: 1

                    debug: 0

                    usex0: 0

                  warning: 1

             cachesolvers: 0

             showprogress: 0

                saveduals: 1

         removeequalities: 0

         savesolveroutput: 0

          savesolverinput: 0

          saveyalmipmodel: 0

        convertconvexquad: 1

    assertgpnonnegativity: 1

             thisisnotagp: 0

                   radius: Inf

                    relax: 0

                  dualize: 0

                savedebug: 0

                   expand: 1

                allowmilp: 1

           allownonconvex: 1

                    shift: 0

                   dimacs: 0

            beeponproblem: [-5 -4 -3 -2 -1]

                bisection: [1x1 struct]

                  bilevel: [1x1 struct]

                   bmibnb: [1x1 struct]

                      bnb: [1x1 struct]

                   cutsdp: [1x1 struct]

                      kkt: [1x1 struct]

                   moment: [1x1 struct]

                       mp: [1x1 struct]

                    mpcvx: [1x1 struct]

                     plot: [1x1 struct]

                   robust: [1x1 struct]

                      sos: [1x1 struct]

                    baron: []

                 bintprog: [1x1 struct]

                   bonmin: []

                     cdcs: [1x1 struct]

                      cdd: [1x1 struct]

                      cbc: [1x1 struct]

                      clp: [1x1 struct]

                    cplex: [1x1 struct]

                     csdp: [1x1 struct]

                     dsdp: [1x1 struct]

                     ecos: []

                 filtersd: [1x1 struct]

                  fmincon: [1x1 struct]

               fminsearch: [1x1 struct]

                    frlib: [1x1 struct]

                     glpk: [1x1 struct]

                   gurobi: [1x1 struct]

                    ipopt: [1x1 struct]

               intlinprog: []

                   knitro: [1x1 struct]

                  linprog: [1x1 struct]

                   lmilab: [1x1 struct]

                  lmirank: [1x1 struct]

                logdetppa: [1x1 struct]

                  lpsolve: [1x1 struct]

                lsqnonneg: [1x1 struct]

                   lsqlin: [1x1 struct]

                     kypd: [1x1 struct]

                      nag: [1x1 struct]

                    mosek: [1x1 struct]

                    nomad: []

                     ooqp: []

                   penbmi: [1x1 struct]

                   penlab: []

                   pensdp: [1x1 struct]

                      pop: [1x1 struct]

                  qpoases: []

                    qsopt: [1x1 struct]

                 quadprog: [1x1 struct]

               quadprogbb: [1x1 struct]

                     scip: []

                      scs: [1x1 struct]

                     sdpa: [1x1 struct]

                    sdplr: [1x1 struct]

                    sdpt3: [1x1 struct]

                   sdpnal: [1x1 struct]

                   sedumi: [1x1 struct]

                sparsepop: [1x1 struct]

               sparsecolo: [1x1 struct]

                     vsdp: [1x1 struct]

                   xpress: []

[Johan Löfberg]

Yes what? You've not turned on debug

[Wanming Hao]

Sorry, how to turn on debug?

[Johan Löfberg]

ops = sdpsettings('debug',1)

[Wanming Hao]

It appers the follows:

??? Error: File: apply_recursive_differentiation.m Line: 43 Column: 7

Expression or statement is incorrect--possibly unbalanced (, {, or [.

Error in ==> fmincon_fun at 78

    dx =

    apply_recursive_differentiation(model,xevaled,requested,model.frecursivederivativeprecompute);

    

Error in ==> fmincon at 528

      [initVals.f,initVals.g(:)] = feval(funfcn{3},X,varargin{:});

Error in ==> callfmincon at 58

[xout,fmin,flag,output,lambda] =

fmincon('fmincon_fun',model.x0,model.A,model.b,model.Aeq,model.beq,model.lb,model.ub,callback_con,model.options.fmincon,model);

Error in ==> solvesdp at 350

    eval(['output = ' solver.call '(interfacedata);']);

Error in ==> Untitled at 40

 sol = solvesdp(F,-Sum,ops)

Caused by:

    Failure in initial user-supplied objective function evaluation. FMINCON cannot continue.

[Johan Löfberg]

The error message looks exactly the same, so the solution is most likely precisely the same

[Wanming Hao]

Your mean that some original codes need to be modified? How to slove it?

Thanks a lot again

[Johan Löfberg]

The error is here

??? Error: File: apply_recursive_differentiation.m Line: 43 Column: 7

Expression or statement is incorrect--possibly unbalanced (, {, or [.

and the reason is once again the use of a ~ as output argument in a function call, which isn't supported on your version of matlab, and hence has to be changed

[Wanming Hao]

Dear John

    Thank you very much for your help. I do it according to your advice and it can work sucessfully. Thanks a lot again again!!

best wishes

[Mark L. Stone]

~ for unused output argument was introduced in MATLAB R2009b http://blogs.mathworks.com/loren/2009/09/11/matlab-release-2009b-best-new-feature-or/ .