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The shakedown theory, which analyses elastic-plastic constructions, subjected by variable repeated load, enables treating moving load as a separate type of variable repeated load. The load is allowed to „move" at any part of the construction: from the middle of the bridge, to turn back, again move ahead -- in this manner loading history is universally evaluated and it is a crucial factor, considering stress-deformation state of structures under plastic deformations. This paper reveals a possibility to apply methods of shakedown theory for creation and solution of theoretical optimization mathematical models of trusses. The perfectly elastic-plastic loaded by moving load truss is considering. The mathematical models of the minimal volume truss or it acting load maximization problems are created. There are evaluating not only strength (shakedown) and rigidity restrictions, but also stability restriction in case of plastic state of truss in models. There is proposed new solution algorithms and introduced numerical examples of truss optimization in case of moving load. The results axe valid for the small displacement assumptions. [ABSTRACT FROM AUTHOR]