Nonlinear operators - integer models

In addition to modeling convex and concave operators and perform automatic analysis and derivation of equivalent conic programs using graph models, YALMIP uses the nonlinear operator framework for implementing logic and combinatorial expression involving commands such as or, and, ne, iff, implies, nnz, alldifferent, sort and ismember, and on a higher level, nonconvex piecewise functions in connection with MPT. The common feature among these operators is that they all require binary and integer variables to be represented in a structured way.

The same framework is used also for alternatives to graph-based implementations. If the convexity propagation of a conic representable function such as min or max fails, thus invalidating the use of graph-models, YALMIP can create an alternative model based on mixed-integer representations. This done for many of the linear programming representable operators.

Mixed-integer representations are also used to model discontinuous functions such as floor, ceil, fix, round, sign, rem, and mod.

Working with mixed-integer representations

Consider the following simple example which violates propagation rules for convexity. YALMIP will detect this, and switch to a mixed-integer representation of the absolute value. The end result is a mixed-integer linear program.

sdpvar x y
F = [abs(abs(x+1)+3) >= y, 0<=x<=3];
sol = optimize(F,-y);
value([x y])
ans =
    3.0000    7.0000

Since the mixed-integer models are based on big-M reformulations, it is crucial that you have explicit bounds on all variables involved in the nonconvex expressions. Read more about this in the big-M tutorial.

If you not want YALMIP to resort to mixed-integer models in nonconvex cases, you can turn off this feature

sdpvar x y
F = [abs(abs(x+1)+3) => y, 0<=x<=3];
sol = optimize(F,-y,sdpsettings('allownonconvex',0));
sol.info

ans =

Convexity check failed (Expected concave function in constraint #1 at level 1)

Mixed-integer model as alternatives

We will start by implementing a rudimentary representation of scalar absolute value, with support for both a graph-model and an integer model. The difference compared to the model we created in [graph-representation] is that we return a mixed-integer model when YALMIP asks for an exact model. The hard part is of course to come up with a suitable integer model. Notice the use of the function derivebounds, which will give us bounds on the argument, thus helping us to obtain a numerically sound big-M model (assuming that explicit bounds have been added to the involved variables in the model)

function varargout = abs(varargin)
switch class(varargin{1})    

    case 'double'
        error('This should have been caught by built-in.')

    case 'char'   
        switch varargin{1}
          case 'graph'
            t = varargin{2};
            X = varargin{3};

            F = [-t <= x <= t];

            properties.convexity    = 'convex';
            properties.monotonicity = 'none';  
            properties.definiteness = 'positive';
            properties.model        = 'graph';	  

            varargout{1} = F;
            varargout{2} = properties;
            varargout{3} = X;

          case 'exact'

            t = varargin{2};
            X = varargin{3};

            [M,m] = derivebounds(X);
            z = binvar(1);
            F = [0 >= x - t >= 2*m*z,
                        m*z <= x,
                 0 <= t + x <= 2*M*(1-z),
                          x <= M*(1-z)];

            properties.convexity    = 'convex';
            properties.monotonicity = 'none';
            properties.definiteness = 'positive';	  
            properties.model        = 'integer';

            varargout{1} = F;
            varargout{2} = properties;
            varargout{3} = X;

          otherwise
            error('Something is very wrong now...')
        end    

    case 'sdpvar' % Always the same for R^n -> R^1
        varargout{1} = yalmip('define',mfilename,varargin{:});    

    otherwise
end

Mixed-integer models as default

Some operators, such as sign, does not have a graph-representation, but must be modelled using an integer representation (or a callback approach).

In these cases, we create an operator that always returns the mixed-integer model, even though YALMIP asks for a graph-model. We communicate the fact that we returned a mixed-integer model via the model field in the properties. By returning an integer model directly instead of simply returning an empty model when YALMIP asks for a graph-model, we reduce the work-load for YALMIP (if YALMIP fails to get a graph model, it will make a second call and ask for an exact model, unless an exact model was returned anyway).

function varargout = sign(varargin)
switch class(varargin{1})    

    case 'double'
        error('This should have been caught by built-in.')

    case 'char'   

        t = varargin{2};
        X = varargin{3};

        d = binvar(1,1);
        [M,m] = derivebounds(X);
        F = [X >= d*m,X <=(1-d)*M, t == 1-2*d];

        properties.convexity    = 'none';
        properties.monotonicity = 'increasing';
        properties.definiteness = 'none';	  
        properties.model        = 'integer';	  

        varargout{1} = F;
        varargout{2} = properties;
        varargout{3} = X;

    case 'sdpvar' % Always the same for R^n -> R^1
        varargout{1} = yalmip('define',mfilename,varargin{:});    

    otherwise
end