Optimization of pure non-sequential (NS) systems is inherently more difficult than optimization of sequential systems. This difficulty stems from defining a merit function and computing the derivative of it with respect to variable parameters. If the energy in a given ray is assigned to only one pixel, there is no quantitative difference when a system change causes the ray to shift anywhere within that pixel. As a result, optimization is difficult, with discontinuous derivatives in the merit function occurring when a ray crosses the boundary into a new pixel; this is a very inefficient method for optimization.
ZEMAX contains several tools that significantly improve the optimization of NS systems. The first is “pixel interpolation” which solves the problem of discontinuous derivatives in the merit function. 
Instead of 100% of a ray’s energy being assigned to the single pixel struck, a fraction of the energy is spread to adjacent pixels based upon the location of the ray intercept inside the pixel. As a result, there is a noticeable change in the merit function as a system change causes a ray to move across a pixel. We will see the difference this makes on optimization efficiency in the following example.
Also, the NSDD optimization operand has received added capabilities: computation of standard deviation and mean of all non-zero pixel values, intensity or irradiance centroid coordinates, and RMS distributions of pixel data from the centroid. For a detailed description of the new NSDD capabilities, see the Optimization chapter in the ZEMAX user’s guide. These features allow more efficient optimization for focus (minimize RMS spatial width), collimation (minimize RMS angular width), and uniformity (best signal to noise).
We will re-optimize the free-form mirror that was used in the article "How to Improve the Brightness of an LED Using a Free-Form Mirror". Download and open the “starting point.zmx” file from the last page of this article. The system modeled contains an LED with a flat mirror in the beam path directing light onto a detector plane. The desire is to optimize the mirror to have the highest degree of beam collimation.
To get an idea of a good place to start the design, we’ll take a look at the universal plot. To open this feature go to Analysis>Universal Plot…New Universal Plot 1D. Enter the settings as shown below: 
Here is a plot of merit function versus base mirror radius without pixel interpolation:
And the following one is with pixel interpolation:
Pixel interpolation removes the discontinuous derivatives in the merit function that can slow down optimization.
For the first system optimization, we will only enable pixel interpolation. Otherwise, the steps and variables will be identical to the previous optimization article; we set more variables for each optimization attempt. We begin with the mirror radius as the only variable and set it to -100. It is usually a good idea to begin with some non-zero radius value. Asking the optimization algorithm to change the radius from infinite to small and finite in one step is very inefficient and can lead to stagnation.
After all variables have been allocated to the optimization, orthogonal descent has come to a solution with a central pixel brightness of 220 Cd. The distribution as shown below has a much lower vergence than optimization without pixel interpolation. 
We will perform the same optimization using instead the added capabilities of the NSDD optimization operand and pixel interpolation. In the previous method for optimizing the mirror, the merit function was defined to increase the brightness (luminous intensity) of the central pixel in angle space. Using this method ZEMAX will attempt to make changes to the mirror that only increase intensity of the central pixel. If a change decreases the vergence of the beam in general, but does not increase central pixel intensity, the optimization algorithm sees the change as ineffectual. However, reduction of beam vergence is clearly beneficial if our goal is to attain a high degree of beam collimation. A better set of operands to use would be centroid position and RMS width of the beam in angular space. We will specify the criteria in the merit function editor:
NSDD operands 6 and 7 specify the target centroid coordinates of the luminous intensity and NSDD operand 8 specifies the target RMS radius of the luminous intensity data. Using these criteria, any change that causes an increase in central pixel luminous intensity will shift the centroid closer to (0,0). Additionally, a reduction in vergence of the beam will cause the RMS radius to move closer to 0. These criteria more accurately describe what we are attempting to achieve and our results will show improved performance as a consequence.
Before we begin to optimize the system, let’s take a look at the universal plot to see how the modified merit function improves optimization efficiency.
The new merit function shows greater continuity even without pixel interpolation. The optimization algorithm can converge on a solution very quickly in such a solution space, and the optimization we perform will confirm this.
In the same manner as previous optimizations, we begin with a radius value of -100. Using this step wise optimization (radius; radius & conic; all coefficients), the DLS algorithm is able to come to a superior design in an astonishing 1/100th of the time it takes the global search optimizing for central pixel luminous intensity. The luminous intensity plot below shows the optimized distribution using the new NSDD operands. 
The central pixel luminous intensity of 260 Cd is higher than that attained with the previous optimization method (~250 Cd), and the RMS width is vastly improved; all in a fraction of the time. If pixel interpolation is disabled for this optimization, the central pixel brightness is even higher (265 Cd) because all the energy hitting the central pixel is assigned to it. A final hammer optimization would further improve this result.
ZEMAX has added several new features that aid in optimizing pure non-sequential systems.
• Pixel interpolation distributes ray energy among adjacent pixels to remove discontinuities in the merit function
• The NSDD operand can also compute the standard deviation and mean of all non-zero pixel data, intensity and irradiance centroid coordinates, and RMS distributions of pixel data from the centroid.