xprsmiqp

Purpose

Solve mixed integer quadratic programming problems with Xpress.

Synopsis

x = xprsmiqp(H,f,A,b,rtype,ctype, clim,sos,lb,ub,x0,options)

[x,fval,exitflag,output] = xprsmiqp(...)

Input arguments

Matrix for quadratic objective terms

Linear objective function vector

Matrix for linear constraints

Vector for constraints RHS

rtype

Character vector (string) giving the row types:

L	indicates a ≤ row
E	indicates a = row
G	indicates a ≥ row
N	indicates a free row

If rtype = [], all rows are assumed to be of type 'L'. If rtype is a single character, all constraints are assigned the corresponding type.

ctype

Character vector (string) giving the column types:

C	(or \0) continuous variables
B	binary variables
I	integer variables
P	partial integer variables
S	semi-continuous variables
R	semi-continuous integers

If ctype = [], all rows are assumed to be of type 'C'. If ctype is a single character, all constraints are assigned the corresponding type.

clim

Vector containing the integer limits for the partial integer variables and lower bounds for semi-continuous and semi-continuous integer variables (column types 'P', 'S', 'R'). Values in the positions corresponding to all other columns are ignored. clim is mandatory if there are any 'P', 'S', or 'R' columns. If clim is a scalar, all columns are assigned to that same limit.

mipstructs

Struct vector defining additional MIP constraints. The number of MIP structs is given by the number of elements in the struct. Each struct must have mipstructs(i).type, which defines the constraint type and the remainder of the struct:

	SOS constraints
mipstructs(i).type='1' or '2'	These define SOS1 or SOS2 constraints, in this case the struct needs to define the following additional vectors:
mipstructs(i).ind	numeric vector with the indices of columns in the set (column indices start from 0);
mipstructs(i).wt	numeric vector with the reference row weights corresponding to the columns in the mipstructs(i).ind vector. It must have the same length as mipstructs(i).ind.
	Indicator constraints
mipstructs(j).type='i'	This declares one of the rows in the constraint matrix as an indicator constraint. In this case the following additional entries are required:
mipstructs(j).row	0-based index of a row in the A-matrix that should be changed to an indicator;
mipstructs(j).col	0-based index of the binary indicator variable that should activate the constraint;
mipstructs(j).comp	Either +1 if the row should be active if the variable takes value one or -1 if the row should be active if the binary takes value zero.
	Piecewise linear constraints
mipstructs(k).type='p'	This declares a piecewise linear constraint y = f(x) and requires the following additional arguments:
mipstructs(k).res	0-based index of the resulting column y;
mipstructs(k).col	0-based index of the input column x;
mipstructs(k).x	numeric vector with x-coordinates of breakpoints that define function f (for more details please refer to the C documentation);
mipstructs(k).y	numeric vector with y-coordinates of breakpoints.
	General constraints
mipstructs(l).type='n' or 'x' or 'd' or 'r' or 's'	This declares a miN, maX, anD, oR or abS general constraint (like y = max (x_i)) with the following additional struct members:
mipstructs(l).res	0-based index of the resulting column y;
mipstructs(l).cols	numeric vector with the indices of the input columns x_i (column indices start from 0); for abs this has to be a single index;
mipstructs(l).vals	numeric vector of constants to include in the min/ max (ignored for and/ or/ abs).

Lower bounds. If lb = [] it means there are no lower bounds. If lb is a scalar, x is uniformly bounded by that scalar.

Upper bounds. If ub = [] it means there are no upper bounds. If ub is a scalar, x is uniformly bounded by that scalar.

Optional initial known solution used to speed-up search.

options

Options structure created with optimset or xprsoptimset functions. See xprsoptimset for more details.

Output arguments

Solution found by the optimization function. If exitflag > 0, then x is a solution; otherwise, x is the value of the optimization routine when it terminated prematurely.

fval

Value of the objective function at the solution x.

exitflag

Integer identifying the reason the optimization algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated.

1	function converged to a solution x (MIPSTATUS=MIP_OPTIMAL)
0	number of iterations exceeded iter limit (STOPSTATUS= STOP_ITERLIMIT)
-2	the problem is infeasible (MIPSTATUS=MIP_INFEAS)
-4	number of searched nodes exceeded limit (STOPSTATUS= STOP_NODELIMIT)
-5	search time exceeded limit (STOPSTATUS= STOP_TIMELIMIT)
-8	other stop reason, see MIPSTATUS and STOPSTATUS for details

output

Structure containing information about the optimization and, eventually, values of Xpress controls and attributes. See quadprog and the Section Setting and querying controls and attributes for details.

Further information

1. This routine finds the minimum of a problem specified by

min 0,5·x'·H·x + f·x
s.t. A·x ≤|=|≥ b
lb≤ x ≤ ub

and x in the domain specified by the ctype, clim and sos arguments, where H is an n × n matrix; A is an m × n matrix; f, b, rtype, ctype, clim, lb, ub, and x0 are vectors; sos is a struct vector.

2. Input arguments rtype and following can be omitted, with the condition that, if one is omitted, also all the following ones must be omitted (as in x=xprsmiqp(H, f, A, b, rtype)). Omitting an input argument has the same effect as passing an empty array [].

3. All output arguments can be omitted too, again with the condition that, if one is omitted, also all the following ones must be omitted (as in [x, fval]= xprsmiqp(H, f, A, b, rtype)).

4. If the specified input bounds for a problem are inconsistent, the output x and fval are set to [].

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