CPLEX and MOSEK with YALMIP gives different results - optimization

Question

I am solving an optimization problem using MATLAB with YALMIP which is similar to a unit commitment problem.
This is a Mixed Integer Linear Programming problem.
I am considering two decision variables --> P and T_room
I formulated the problem in MATLAB with YALMIP,
When I use CPLEX and MOSEK solvers, I am getting different results.
Can anyone explain to me why that is happening?
CPLEX solver results
MOSEK solver results
clc;
clear all;
rng(0,'twister');
%% time step of the demand response --> 15 mins
deltaT = 15/60 ;
%% number of air conditioners
Nunits = 3;
%% demand required for each interval
Pdemand = [2 3 4 2 5 5 8 9]';
% Pdemand = randi([2,10],8,1);
%% time interval considered for the study --> 15 mins intervals for 2
%%% hours
Horizon = length(Pdemand) ;
%% cost for each appliance ( need to think little about this approach )
Q = diag([rand(1,Nunits)]);
%% defining the possible reduction levels for the appliances
% ratedPower = [1 6 4]';
ratedPower = randi([5,7],Nunits,1);
%% defining minimum and maximum power limits for the appliances
Pmax = randi([6,8],Nunits,1);
Pmin = zeros(Nunits,1);
% possible power levels for appliances (discrete levels)
LevelsOutput = [0 0.5 0.75 1.0] ;
% matrix containing all the possible levels of output power for all the
%% appliances
PowerLevelMatrix = ratedPower * LevelsOutput ;
%% parameters based on the temperature
%%% outside temperature (we assume that the outdoor temperature remains the same throughout the optimisation process)
T_outdoor = randi([34, 38], Nunits,1);
%%% preferred lower level of temperature for the appliances
T_lower = randi([20,21], Nunits,1) ;
%%% preferred upper level of temperature for the appliances
T_upper = randi([28, 29], Nunits,1) ;
%%% initial temperature for the appliances
T_initial = randi([22, 28], Nunits,1);
%% parameters of the house
%% equivalent thermal resistance
R = (2.5-1.5)*rand(Nunits,1) + 1.5 ;
%%% equivalent thermal capacitance
Ca = (2.5-1.5)*rand(Nunits,1) + 1.5 ;
%% for the air condtioner
c = repmat(0.03,Nunits,1);
d = repmat(-0.4,Nunits,1);
a = repmat(0.06,Nunits,1);
b = repmat(-0.3,Nunits,1);
%% defining the variables of the demand response problem
%%% power levels for the appliances
P = sdpvar(Nunits, Horizon, 'full');
%% slack variable for exceeding the demand
% Demand_Penalty = 50;
% DemandSlack = sdpvar(1,Horizon);
%% decision variable for the room temperature
T_room = sdpvar(Nunits,Horizon,'full') ;
%% Constraints for the optimisation problem
Constraints = [];
for k = 1: Horizon
% Constraints = [Constraints, sum(P(:,k)) + DemandSlack(k) >= Pdemand(k)];
% Constraints = [Constraints,DemandSlack(k) >= 0];
Constraints = [Constraints, sum(P(:,k))>= Pdemand(k)];
end
%%% constraint on min amd max power levels
for k = 1: Horizon
Constraints = [Constraints, Pmin <= P(:,k) <= Pmax];
end
%%% adding the constraint on discrete power levels for each appliance
for nApp = 1 : Nunits
for k = 1 : Horizon
Constraints = [Constraints, ismember(P(nApp,k),PowerLevelMatrix(nApp,:))];
end
end
%% the temperature should be within the desired limits --> cannot go beyond those limits
for k = 2: Horizon
Constraints = [Constraints, T_lower <= T_room(:,k) <= T_upper ];
end
%% adding the initial constraint of temperature
Constraints = [Constraints, T_room(:,1) == T_initial];
%%% adding the temperature model of the constraint for the inverter type
%%% air condtioner
for k = 2: Horizon
Constraints = [Constraints, T_room(:,k) == T_outdoor - ((a./c).*P(:,k) + ((b.*c - a.*d)./c)).*R.*(1- exp((-deltaT)./(R.*Ca))) - T_outdoor.* exp((-deltaT)./(R.*Ca)) + T_room(:,k-1).* exp((-deltaT)./(R.*Ca)) ];
end
%% adding a sample constraints to the temperature update
% for k =2: Horizon
%
% Constraints = [Constraints, T_room(:,k) == T_room(:,k-1) ];
% end
%%% defining the objective function
% weighting factors
alpha = 1 ;
beta = 1 ;
Objective = 0 ;
for k = 2 : Horizon
Objective = Objective + alpha * Q * P(:,k) + beta * ((2* T_room(:,k) - T_lower - T_upper)./(T_upper - T_lower));
% Objective = Objective + P(:,k)' * Q * P(:,k);
% Objective = Objective + Demand_Penalty * DemandSlack(k) ;
end
%%% solving the optimisation problem
%
ops = sdpsettings('verbose',2, 'debug', 1, 'solver', 'mosek');
% ops = sdpsettings('verbose',1, 'debug', 1);
solution = optimize (Constraints, Objective, ops) ;
Pout = value(P);
T_room_out = value(T_room);

Answer 1

Your code does not make sense to begin with, as your objective is a vector
>> Objective
Linear matrix variable 3x1 (full, real, 42 variables)
With that, YALMIP will solve three optimization problems, and then values you obtain when applying value command will be the values from the last solution.
Having said that, concentrating on the last objective you have defined (although I think you intended to minimize the sum or something), I get the same objective with both solvers (not the same solution, the solution simply isn't unique)
>> solution = optimize (Constraints, Objective(3), sdpsettings('solver','cplex'))
>> value(Objective(3))
ans =
1.8418
>> value(P)
ans =
0 5.2500 3.5000 5.2500 3.5000 5.2500 5.2500 5.2500
0 3.0000 3.0000 6.0000 6.0000 6.0000 3.0000 4.5000
5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 0 0
>> solution = optimize (Constraints, Objective(3), sdpsettings('solver','mosek'))
>> value(Objective(3))
ans =
1.8418
ans =
3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 5.2500 5.2500
3.0000 3.0000 3.0000 3.0000 6.0000 0 3.0000 6.0000
3.7500 5.0000 5.0000 5.0000 5.0000 5.0000 0 0