24 May 2022
Chen et al(2021) suggest that a optimization solver may be hard to obtain good enough solutions a Day-Ahead(DA) SCUC problem
(as large as 50k binary variables and 15k transmission constraints) within available time.
Thus, efforts are made from both sides: faster solvers and more efficient ad-hoc algorithms.
Since the first release of the mathematical programming package CPLEX in 1988, the advancement in algorithm,
along with the improvement in hardware, has speeded up the problem solving by 5.28 million times till 2004 (Bixby, 2012).
That has enabled more and more industries to embrace optimization solvers, including power grids.
From the recent benchmarking results of Mittlemann (2022), the fastest commercial LP solver runs more than 90 times faster than
a popular open-source one, and 20 times faster for MILP solvers.
- Bixby, R. E. (2012). A brief history of linear and mixed-integer programming computation. Documenta Mathematica, 2012, 107-121.
- Mitllemann, H. (2022). http://plato.asu.edu/ftp/lpsimp.html (retrieved on 24 May, 2022)
- Chen, Y., Pan, F., Holzer J., Rothberg E., Ma, Y. and Veeramany, A. (2021) A high performance computing based market economics driven
neighborhood search and polishing algorithm for security constrained unit commitment. IEEE Transactions on Power Systems, 36(1), 292-302
17 Dec 2015
The term, physical modeling, refers to the activity identifying unknown physical relationship of/within the system of interest.
When we describe satellites’ motion in Newton mechanics, we accept the absolute time-space assumptions implicitly. However, wrong results are obtained. We can establish a model using mathematical languages, enumerating all (obviously) known relations of the the system straightforwardly.
But, can any behavior that should be considered can be derived from this model?
This question looks like to ask can the * relativity * be derived from Newton mechanics?
31 Oct 2015
For extremely complicated optimization problems, mathematical methods usually fail in finding the exact optima. Smart researchers resort to approximations, which are sometimes based on experience or intuition, and often unexplainable in rigorously mathematical ways. However, they may perform pretty well in practice.
To find, or to develop, approximations for extremely complicated optimization problems, we can not rely only on known mathematics, which provides us specific points of view. What if there is any point of view, or notion, that is unknown by far but actually would lead us to brand new mathematics?
If we compare the known mathematics to the Geocentrism (Geocentric Model), they do provide some descriptions and clues about planets, but in an extremely complicated way. However, when it comes to Heliocentrism, the orbiting of planets seems to be much simpler.
We are probably able to state that complication may be a lack of appropriate model. A scientific way is to observe, guess, experiment and revise. By repeating this iteration enough times, a much more accurate new model for the problem of interest is expected to form. However, the resulted model may be unable to be totally explained mathematically. So, it might be the mathematics’ fault, isn’t it?
The equation, $E=mc^{2}$, seems to be absolute nonsense in mathematical meaning for a person doesn’t know the physical relationship.
If we develop a mathematical model for a physical system, it usually means that we have enough knowledge about the system. This is plausible for relatively small problems with straightforward mechanisms. However, for large complicated system, an appropriate mathematical model can be never developed without looking into the physical system itself.
An optimization problem can be formulated as objective function(s) and the valid domain, e.g. $\max f(x,y)= \sin(x+\max{|x|, y})$, where $x\in X, y\in Y$. A typical search procedure can be described as the following:
-
Select an initial point ( $x_{0}, y_{0}$ ) in the domain
-
Proceed to next point according to a direction, or to stop
-
Repeat the above two steps (until stopped)
Obviously, the initial point selection and directions of each proceeding are critical to the performance of the procedure. In special cases, e.g., linear or convex, there are mature procedures and guidelines, e.g., the simplex method.
The term Meta-Strategy is used to refer to a method for a strategy, i.e., a rule or procedure that generates outcomes based on available information and knowledge.
31 Oct 2015
For the scheduling in production, the NP-hardness of the problems demands efficient and effective approximations. However, approximations for general problem settings may perform worse than specially designed ones for the specific settings. But as the reconfiguration of production flows becomes more and more frequent in response to varying and fluctuating market demands, and the production environment itself is actually dynamic in nature as well, the new settings also need to re-configure even to re-design, frequently.
…
