Degenerate optimization problems of economics and power engineering
Optimization of large economic and power engineering systems leads to degenerate solutions of high dimension. This is a very strong mathematical complication. However it allows to consider future development of the power industry based on simultaneous use of nuclear power plants (NPPs) together with coal- and gas power plants, or solely on NPPs. This requires system optimization of NPP parameters.
Calculations of optimal systems of high dimension have shown that the degenerate space of possible solutions for economics and power engineering can be seen as a set of points on a lunar surface pitted with a finite number of craters. This degenerate space can be referred to as «non-convex, non-concave».
In other words, the N-dimensional degenerate «non-convex, non-concave» space of large dimension (N ≥ 10 000) resembles a «lunar surface» with craters of different depth. Craters are the neighborhoods of locally-optimal solutions, the latter being at the crater’s bottom. The crater’s depth defines the value of the optimization functional. Among the most deep, but otherwise different craters, there are craters of equal depth, i.e. craters with the same value of the locally-optimized optimization functional. Local optima (local plans) in different craters can differ by the structure but have the same value of the optimization functional.
Calculations show that among the craters of equal size (with the same values of the functional at the locally-optimized plans of economics and power engineering development) there are craters with locally-optimal plans of development (among other possible heterogeneous combinations of economics and power technologies) only with coal and gas, or coal, gas and nuclear facilities, or only with NPPs. Comparing the values of the optimization functional in different craters, one can find the optimal solution – a locally-optimal plan with the best value of the functional (for example, in the case of minimization – with the minimal value of the functional among all craters).
- Linejnye neraventstva i smezhnye voprosy. Sbornik statej pod redakciej H.W. Kuhn i A.W. Tucker [Linear Inequalities and Related Systems. Edited by H.W. Kuhn and A.W. Tucker. Princeton. 1956]. Moscow, Inostrannaya Literatura Publ., 1959. 472 p. (in Russian).
- Gass S.I. Linejnoe programmirovanie (metody i prilozheniya) [ Gass S.I. Linear Programming. Methods and Applications. New York-Toronto-London. 1958]. Moscow, Fiziko-matematicheskoj Literatury State Publ., 1961. 304 p. (in Russian).
- Dantzig G.B. Linejnoe programmirovanie, ego obobscheniya i primeneniya [Dantzig G.B. Linear Programming and Extensions. Prinseton-New Jersey. 1963]. Moscow, Progress Publ., 1966. 600 p. (in Russian).
- Intriligator M. Matematicheskie metody optimizatsii i ekonomicheskaya teoriya [Intriligator M. Mathematical Optimization and Economic Theory. N.Y., 1971]. Moscow, Progress Publ., 1971. 606 p. (in Russian).
- Klimenko A.V. Mozhet li yadernaya energetika stat’ konkurentosposobnoj na svobodnom rynke energii [There can be a nuclear power to competitive energy in the free market]. Izvestiya vuzov. Yadernaya energetika. 2013, no. 4, pp. 17-28. (in Russian).