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algorithms analysis equations applied approximations assumed basic feasible solution behavior binomial series calculated compatibility conditions compressive stresses computational effort considered convergence convex convex programming coordinates corresponding cross sections cross-sectional areas defined derivatives design procedures design space design variables direction displacement constraints efficient elements equality constraints equilibrium example feasible design feasible region geometrical variables geometry given homogeneous functions inequality constraints initial design iteration linear programming loading conditions lower bound LP problem LP solution minimum modified modulus of elasticity nonlinear programming nonnegative number of variables objective function obtained optimal design problem optimal solution optimal topologies optimality criteria optimization methods optimization problem penalty function plastic plastic analysis problem formulation quadratic quadratic function redundant forces satisfied scaling shown in Fig solution process solved statically determinate statically determinate structure stiffness matrix stress constraints structural design structural optimization subproblems Substituting Taylor series Taylor series expansion truss shown vector