Final Remarks

Computational complexity of MP-problems depends heavily upon their degree of non-linearity. It is quite easy to solve large LP-problems. Several simplex codes available on the market are able to solve problems with hundreds of thousands of variables and/or constraints. These codes usually use some version of sparse-matrix technique in order to cope with large matrices. Interestingly enough, the simplex algorithm, discovered over 50 years ago, is still the best solver.

QP-problems are more demanding and one can hardly expect to solve in reasonable time a problem with more than couple of hundreds variables and/or constraints. Despite huge effort spend on developing general purpose Non-Linear Programming solvers, the result is rather unsatisfactory. Most available codes work sufficiently well in the range of several tenths of variables and/or constraints.

Structural analysis and optimization taking into account elastic properties of the material is not reducible to Linear Programming. On the other hand, the efficiency of QP – and NLP-solvers is far below the efficiency of modern solvers of the sets of linear algebraic equations. This explains why expectations that Mathematical Programming will replace Linear Algebra in the domain of computing were not met.

On the other hand, the language of Mathematical Programming is excellent in teaching Structural Analysis and Structural Optimum Design. It discloses common background of the broad class of problems governed by geometrically linear kinematics, allows students to grasp the principal difference between bilaterally and unilaterally constrained problems, trains them in a good custom of looking at each problem from two perspectives – the kinematic one and the static one, simplifies checking of existence and uniqueness of solutions.

Obviously, a prerequisite of teaching the MP-based approach to the theory of structures is the prior knowledge of the Mathematical Programming by the students. A class on this subject should be taught during the first or second year of undergraduate studies, as a supplement to courses on Linear Algebra and Differential Calculus. The knowledge acquired on the MP-theory could be exploited in teaching not only Structural Analysis and Structural Optimum Design but also in the classes on other aspects of Civil Engineering (e. g. road planning, cost optimization, construction planning, etc.).


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