Special Features in Constrained Optimization MT¶

The following sections describe the special features found in Constrained Optimization MT.

Structures¶

In COMT, the same procedure that computes the objective function will also be used to compute analytical derivatives if they are being provided. This procedure will have an additional argument which tells the function whether to compute the log-likelihood or objective, the first derivatives, the second derivatives, or all three. This means that calculations in common will not have to be redone.

`modelResults` Structure¶

This objective procedure will return a modelResults structure which has three member variables:

function: Scalar value of the objective function.
gradient: Optional Kx1 vector of first derivatives.
Hessian: Optional KxK matrix of second derivatives.

// Example objective function
proc (1) = myobjective(parms, ind);
    struct modelResults mm;

    // Perform any calculations common to
    // objective function, gradient, and Hessian

    // If the first element of 'ind' is
    // non-zero, calculate objective function
    if ind[1];
      mm.function = // Calculate objective function
    endif;

    // If the second element of 'ind' is
    // non-zero, calculate gradient
    if ind[2];
      mm.gradient = // Calculate gradient
    endif;

    // If the third element of 'ind' is
    // non-zero, calculate Hessian
    if ind[3];
      mm.Hessian = // Calculate Hessian
    endif;

    // Return modelResults structure
    retp(mm);
endp;

In the objective function the function value return is required. However, the derivatives are optional or even partially optional, i.e., you can compute a subset of the derivatives if you like, and the remaining will be computed numerically. When computing only a subset of the derivatives, set the uncomputed element of the gradient vector to a missing value. COMT will attempt to compute numerical derivatives for any element of the gradient vector that contains a missing value.

Parameter Vector (PV) Structure¶

COMT allows you to use the PV structure from the standard GAUSS Run-Time Library to pass parameters to the objective function. The PV structure makes it easy to store your parameters as vectors, matrices, or n-dimensional arrays. For cases in which your parameter vector is simply a vector, COMT allows you to pass in your parameter vector directly without the use of the PV structure.

// Add symmetric matrix of starting
// values to 'PV' structure
omega_strt = {  1.0 0.8 -0.4,
                0.8 1.0  0.6,
               -0.4 0.6  1.0 };
p = pvPackS(pvCreate(), omega_strt, "omega");

proc (1) = myobjective(struct PV parms, ind);
local omega;

// Retrieve updated symmetric matrix
// inside of objective function
omega = pvUnpack(parms, "omega");

// Perform calculations and return

No more do you have to struggle to get the parameter vector into matrices for calculating the function and its derivatives, trying to remember or figure out which parameter is where in the vector. If your log-likelihood uses matrices or arrays, you can store them directly into the PV structure and remove them as matrices or arrays with the parameters already plugged into them. The PV structure can even efficiently handle symmetric matrices where parameters below the diagonal are repeated above the diagonal.

The functions pvPackM() and pvPackMI() allow you to specify some elements inside your PV structure as fixed values and others as free parameters. It remembers the fixed values and only updates the values of the free parameters.

Optional Dynamic Arguments¶

There will no longer be any need to use global variables. Any inputs that your procedure needs other than the parameters of the model can be passed into COMT as optional dynamic arguments. These optional arguments will be passed directly and untouched to your objective function.

// Inputs to objective function for
// COMT version 1.0
proc (1) = myobjective(struct PV parms, struct DS d, ind);

// Inputs to objective function for
// COMT current version that requires no
// data other than model parameters.
// And the parameters are simply a vector.
proc (1) = myobjective(x, ind);

// Inputs to objective function for
// COMT current version that requires no
// data other than model parameters.
// And the parameters are packed in a PV struct.
proc (1) = myobjective(struct PV parms, ind

// Inputs to objective function for
// COMT current version that requires
// 2 extra matrices 'theta' and 'gamma'
// Place extra inputs between the parameter vector and 'ind'
proc (1) = myobjective(x, theta, gamma, ind);

// Inputs to objective function for
// COMT current version that requires
// 2 extra matrices 'theta' and 'gamma'
// and using the PV structure for parameters
// Place extra inputs between 'PV' struct and 'ind'
proc (1) = myobjective(struct PV parms, theta, gamma, ind);

Previous versions of COMT required the use of the DS structure for this purpose. The current version is backwards compatible with version 1 so programs written using the DS structure will continue to work.

Control Structures¶

The functions in this library also use control structures to set optimization options rather than global control variables. This means in addition to thread safety that it will be straightforward to nest calls to COMT inside of a call to COMT or other multi-threaded GAUSS functions.

// Declare 'c0' to be a comtControl struct
struct comtControl c0;

// Fill 'c0' with default settings
c0 = comtControlCreate();

// Turn on threading of numerical derivatives in COMT
c0.useThreads = 1;

An important advantage of threading occurs in computing numerical derivatives. If the derivatives are computed numerically, threading will significantly decrease the time of computation.

Threading¶

If you have a multi-core processor in your computer, you may take advantage of this capability by selecting threading. This is done by setting the useThreads member of the comtControl instance.

Augmented Lagrangian Penalty Line Search Method¶

An augmented Lagrangian penalty method with second-order correction described by Conn, Gould, and Toint (2000) Section 15.3.1 is implemented in COMT.

// Example usage of Augmented Lagrangian Penalty Line Search Method
struct comtControl ctl = comtControlCreate();
ctl.algorithm = 1; // Use a specific algorithm
// Additional configuration here

This method requires that constraints be imposed on the parameters. This method is particularly useful in certain optimization scenarios and is fully supported within COMT.

Getting Started

The Objective Function