Directional Necessary Optimality Conditions for Bilevel Programs

The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian–Fromovitz constraint qualification (MFCQ). In this paper, we show that even the first order sufficient condition for metric subregularity (which is, in general, weaker than MFCQ) fails at each feasible point of the bilevel program. We introduce the concept of a directional calmness condition and show that, under the directional calmness condition, the directional necessary optimality condition holds. Although the directional optimality condition is, in general, sharper than the nondirectional one, the directional calmness condition is, in general, weaker than the classical calmness condition and, hence, is more likely to hold. We perform the directional sensitivity analysis of the value function and propose the directional quasi-normality as a sufficient condition for the directional calmness. An example is given to show that the directional quasi-normality condition may hold for the bilevel program.

APPLICATION OF GRADIENT PROJECTION METHOD TO PARAMETRIC OPTIMIZATION OF STEEL LATTICE PORTAL FRAME

The paper has proposed a mathematical model for parametric optimization problem of the steel lattice portal frame. The design variable vector includes geometrical parameters of the structure (node coordinates), as well as cross-sectional dimensions of the structural members. The system of constraints covers load-carrying capacities constraints formulated for all design sections of structural members of the steel structure subjected to all ultimate load case combinations. The displacements constraints formulated for the specifiednodes of the steel structure subjected to all serviceability load case combinations have been also included into the system of constraints. Additional requirements in the form of constraints on lower and upper values of the design variables, constraints on permissible minimal thicknesses, constraints on permissible maximum diameter-to-thickness ratio for the structural members with circle hollow sections, as well as the conditions for designing gusset-less welded joints between structural members with circle hollow sections have been also considered in the scope of the mathematical model. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used to solve the formulated parametric optimization problem. New optimal layouts of the steel lattice portal frame by the criterion of the minimum weight, as well as minimum costs on manufacturing and erection have been presented.

Size optimization of single edge folds for cold-formed structural members

Parametric optimization problem for single edge fold size in cold-formed structural members subjected to central compression has been considered by the paper. Determination the load-bearing capacity of the cold-formed structural members has been performed using the geometrical properties calculated based on the constructed “effective” (reduced) cross-sections taking into account local buckling effects in the section as well as distortional buckling effects. Single edge fold size in cold-formed C-profile has been considered as design variable. Linear convolution of criteria, namely minimization criterion of design area of stiffener cross-section and maximization criterion effective area of stiffener cross-section which defines it reduced load-bearing capacity due to flexural buckling has been used as optimization criterion. The parametric optimization problem has been solved using the method of objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations. In order to realize the formulated optimization problem, software OptCAD intended to solve parametric optimization problems for steel structural systems has been used. Optimization results of the single edge folds for the cold-formed С-profiles manufactured by «Blachy Pruszyński» company, «BF FACTORY» company as well as «STEELCO» company have been presented by the paper. The results of the performed investigation can be used as recommendations for companies-manufacturers of the cold-formed profiles, as well as a guide for creation the national assortment base of the effective cold-formed profiles promoting wider implementation of cold-formed steel structures in building practice.

Optimal energy management of microgrid based on multi-parameter dynamic programming

With the wide application of microgrid system, fluctuation and randomness are the characteristics of distributed generation output. The traditional energy management system can’t meet the requirements to ensure the security and stability of the grid. The microgrid energy management is of great significance to the stable operation of power grid. In order to obtain higher economic benefits and pay less environmental costs, reasonable scheduling of various distributed power sources is able to achieve this goal. In this article, microgrid energy management including distributed generation is studied. The objective function includes the economic objective and the environmental objective. The model of energy management is considered as a multi-objectives and multi-parametric optimization problem. The multi-parameter dynamic programming is used to optimize the energy management of microgrid. Finally, the efficiency of the proposed method is examined by the simulation studies.

Parametric Optimization for Structural Design Problems

Parametric Optimization is used to solve problems that have certain design variables as implicit functions of some independent input parameters. The optimal solutions and optimal objective function values are provided as functions of the input parameters for the entire parameter space of interest. Since exact solutions are available merely for parametric optimization problems that are linear or convex-quadratic, general non-convex non-linear problems require approximations. In the present work, we apply three parametric optimization algorithms to solve a case study of a benchmark structural design problem. The algorithms first approximate the nonlinear constraint(s) and then solve the optimization problem. The accuracy of their results and their computational performance are then compared to identify a suitable algorithm for structural design applications. Using the identified method, sizing optimization of a truss structure for varying load conditions such as a varying load direction is considered and solved as a parametric optimization problem to evaluate the performance of the identified algorithm. The results are also compared with non-parametric optimization to assess the accuracy of the solution and computational performance of the two methods.

Lyapunov Matrices Approach to the Parametric Optimization of a System with Two Delays

In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with two commensurate delays and a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of the Lyapunov functional for the initial function of time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix.

Lyapunov matrices approach to the parametric optimization of a neutral system

In the paper a Lyapunov matrices approach to the parametric optimization problem of a neutral system with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the neutral system. The Lyapunov functional is determined by means of the Lyapunov matrix.

Lyapunov matrices approach to the parametric optimization of time-delay systems

In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix