qabel.gms : Linear Quadratic Control Problem

**Description**

This is a QCP formulation of the original ABEL model. Note that this model is convex and should be very easy to solve. The Linear Quadratic Riccati Equations are solved as a QCP. Nonlinear Programming Problem instead of the usual Matrix QMax can be easily extended to 1000, that allows to pass larger QPs to the solver.

**Reference**

- Kendrick, D, Caution and Probing in a Macroeconomic Model. Journal of Economic Dynamics and Control 4, 1 (1982), 149-170.

**Small Model of Type :** QCP

**Category :** GAMS Model library

**Main file :** qabel.gms

```
$title Linear Quadratic Control Problem as QCP (QABEL,SEQ=293)
$onText
This is a QCP formulation of the original ABEL model. Note
that this model is convex and should be very easy to solve.
The Linear Quadratic Riccati Equations are solved as a QCP.
Nonlinear Programming Problem instead of the usual Matrix
QMax can be easily extended to 1000, that allows to pass
larger QPs to the solver.
Kendrick, D, Caution and Probing in a Macroeconomic Model. Journal of
Economic Dynamics and Control 4, 2 (1982).
Keywords: quadratic constraint programming, Riccati equations, macro economics,
fiscal policy, convex optimization
$offText
$if not set qmax $set qmax 75
Set
n 'states' / consumpt, invest /
m 'controls' / gov-expend, money /
k 'quarters' / q1*q%qmax% /
ku(k) 'control horizon'
ki(k) 'initial period'
kt(k) 'terminal period';
Alias (n,np), (m,mp);
ku(k) = ord(k) < card(k);
ki(k) = ord(k) = 1;
kt(k) = not ku(k);
Table a(n,np) 'state vector matrix'
consumpt invest
consumpt .914 -.016
invest .097 .424;
Table b(n,m) 'control vector matrix'
gov-expend money
consumpt .305 .424
invest -.101 1.459;
Table wk(n,np) 'penalty matrix for states - input'
consumpt invest
consumpt .0625
invest 1;
Table lambda(m,mp) 'penalty matrix for controls'
gov-expend money
gov-expend 1
money .444;
Parameter
c(n) 'constant term' / consumpt -59.4, invest -184.7 /
xinit(n) 'initial value' / consumpt 387.9, invest 85.3 /
uinit(m) 'initial controls' / gov-expend 110.5, money 147.1 /
xtilde(n,k) 'desired path for x'
utilde(m,k) 'desired path for u'
w(n,np,k) 'penalty matrix on states';
w(n,np,ku) = wk(n,np);
w(n,np,kt) = 100*wk(n,np);
xtilde(n,k) = xinit(n)*power(1.0075,ord(k) - 1);
utilde(m,k) = uinit(m)*power(1.0075,ord(k) - 1);
* display w, xtilde, utilde;
Variable
x(n,k) 'state variable'
u(m,k) 'control variable'
j 'criterion';
Equation
criterion 'criterion definition'
stateq(n,k) 'state equation';
criterion..
j =e= .5*sum((k,n,np), (x(n,k) - xtilde(n,k))*w(n,np,k)*(x(np,k) - xtilde(np,k)))
+ .5*sum((ku,m,mp),(u(m,ku) - utilde(m,ku))*lambda(m,mp)*(u(mp,ku) - utilde(mp,ku)));
stateq(n,k+1)..
x(n,k+1) =e= sum(np, a(n,np)*x(np,k)) + sum(m, b(n,m)*u(m,k)) + c(n);
Model abel / all /;
x.l(n,k) = xinit(n);
u.l(m,k) = uinit(m);
x.fx(n,ki) = xinit(n);
option limCol = 0, limRow = 0, solPrint = off;
solve abel minimizing j using qcp;
display x.l, u.l;
```