Exactness of penalization for exact minimax penalty function method in nonconvex programming

The exact minimax penalty function method is used to solve a noncon- vex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exact minimax penalty function method are established by assuming that the functions constituting the considered con- strained optimization problem are invex with respect to the same function η （with the exception of those equality constraints for which the associated Lagrange multipliers are negative these functions should be assumed to be incave with respect to η）. Thus, a threshold of the penalty parameter is given such that, for all penalty parameters exceeding this threshold, equivalence holds between the set of optimal solutions in the considered constrained optimization problem and the set of minimizer in its associated penalized problem with an exact minimax penalty function. It is shown that coercivity is not suf- ficient to prove the results.

THE CONVERGENCE OF APPROACH PENALTY FUNCTION METHOD FOR APPROXIMATE BILEVEL PROGRAMMING PROBLEM

In this paper, a new algorithm-approximate penalty function method is designed, which can be used to solve a bilevel optimization problem with linear constrained function. In this kind of bilevel optimization problem. the evaluation of the objective function is very difficult, so that only their approximate values can be obtained. This algorithm is obtained by combining penalty function method and approximation in bilevel programming. The presented algorithm is completely different from existing methods. That convergence for this algorithm is proved.

APPLICATION OF PENALTY FUNCTION METHOD IN ISOPARANIETRIC HYBRID FINITE ELEMENT ANALYSIS

By the aid of the penalty function method, the equilibrium restriction conditions were introduced to the isoparametric hybrid finite element analysis, and the concrete application course of the penalty function method in three-dimensional isoparametdc hybrid finite element was discussed. The separated penalty parameters method and the optimal hybrid element model with penalty balance were also presented. The penalty balance method can effectively refrain the parasitical stress on the premise of no additional degrees of freedom. The numeric experiment shows that the presented element not only is effective in improving greatly the numeric calculation precision of distorted grids but also has the universality.

A PENALTY FUNCTION METHOD FOR THE PRINCIPAL-AGENT PROBLEM WITH AN INFINITE NUMBER OF INCENTIVE-COMPATIBILITY CONSTRAINTS UNDER MORAL HAZARD

In this paper,we propose an iterative algorithm to find the optimal incentive mechanism for the principal-agent problem under moral hazard where the number of agent action profiles is infinite,and where there are an infinite number of results that can be observed by the principal.This principal-agent problem has an infinite number of incentive-compatibility constraints,and we transform it into an optimization problem with an infinite number of constraints called a semi-infinite programming problem.We then propose an exterior penalty function method to find the optimal solution to this semi-infinite programming and illustrate the convergence of this algorithm.By analyzing the optimal solution obtained by the proposed penalty function method,we can obtain the optimal incentive mechanism for the principal-agent problem with an infinite number of incentive-compatibility constraints under moral hazard.

PENALTY FUNCTION METHOD OF CONTINUUM SHAPE OPTIMIZAION

The penalty function method of continuum shape optimization and its sensitivity analysis technique are presented. A relatively simple integrated shape optimization system is developed and used to optimize the design of the inner frame shape of a three-axis test table. The result shows that the method converges well, and the system is stable and reliable.

Method for electromagnetic detection satellites scheduling based on genetic algorithm with alterable penalty coefficient

The electromagnetic detection satellite （EDS） is a type of earth observation satellites （EOSs）. The Information collected by EDSs plays an important role in some fields, such as industry, science and military. The scheduling of EDSs is a complex combinatorial optimization problem. Current research mainly focuses on the scheduling of imaging satellites and SAR satellites, but little work has been done on the scheduling of EDSs for its specific characteristics. A multi-satellite scheduling model is established, in which the specific constrains of EDSs are considered, then a scheduling algorithm based on the genetic algorithm （GA） is proposed. To deal with the specific constrains of EDSs, a penalty function method is introduced. However, it is hard to determine the appropriate penalty coefficient in the penalty function. Therefore, an adaptive adjustment mechanism of the penalty coefficient is designed to solve the problem, as well as improve the scheduling results. Experimental results are used to demonstrate the correctness and practicability of the proposed scheduling algorithm.

Dynamic Analysis of A Deepwater Drilling Riser with A New Hang-off System

The safety of risers in hang-off states is a vital challenge in offshore oil and gas engineering.A new hang-off system installed on top of risers is proposed for improving the security of risers.This approach leads to a challenging problem:coupling the dynamics of risers with a new hang-off system combined with multiple structures and complex constraints.To accurately analyze the dynamic responses of the coupled system,a coupled dynamic model is established based on the Euler-Bernoulli beam-column theory and penalty function method.A comprehensive analysis method is proposed for coupled dynamic analysis by combining the finite element method and the Newmarkβmethod.An analysis program is also developed in MATLAB for dynamic simulation.The simulation results show that the dynamic performances of the risers at the top part are significantly improved by the new hang-off system,especially the novel design,which includes the centralizer and articulation joint.The bending moment and lateral deformation of the risers at the top part decrease,while the hang-off joint experiences a great bending moment at the bottom of the lateral restraint area which requires particular attention in design and application.The platform navigation speed range under the safety limits of risers expands with the new hang-off system in use.

An improved local radial point interpolation method for transient heat conduction analysis

The smoothing thin plate spline （STPS） interpolation using the penalty function method according to the optimization theory is presented to deal with transient heat conduction problems. The smooth conditions of the shape functions and derivatives can be satisfied so that the distortions hardly occur. Local weak forms are developed using the weighted residual method locally from the partial differential equations of the transient heat conduction. Here the Heaviside step function is used as the test function in each sub-domain to avoid the need for a domain integral. Essential boundary conditions can be implemented like the finite element method （FEM） as the shape functions possess the Kronecker delta property. The traditional two-point difference method is selected for the time discretization scheme. Three selected numerical examples are presented in this paper to demonstrate the availability and accuracy of the present approach comparing with the traditional thin plate spline （TPS） radial basis functions.

An Exact Penalty Function Method for Linear Static Stackelberg Problem with Multiple Leaders and Multiple Followers

The paper models and analyzes the linear static stackelberg model with multiple leaders and multiple followers in which each decision maker within his group may or may not cooperate. An exact penalty function method is developed. The duality gaps of the followers’ problems are appended to the leaders’ objective function with a penalty. This structure leads to the decomposition of the composite problem into a series of linear programming leading to an efficient algorithm. We prove that local optimality is reached for an exact penalty function and illustrste the method with three examples. Our model extends the stackelberg leader-follower model. The analysis in this paper hopefully provides some forther insights about types of the static stackelberg leader-follower model.

Research on optimization of valve open time of the launch barge's ballast tanks

Launch barge is an effective tool for transporting ship segments from one place to another in shipyards. During shifting of segments onto a barge, the slideway on the barge＇s deck must be adjusted to maintain the same level as the wharf and also the barge must be kept level by adjusting the water in the ballast tanks. When to open the adjusting valves is an important factor influencing the barge＇s trim during the water-adjustment process. Because these adjustments are complex a mathematical model was formulated,after analyzing the characteristics of the process of moving the segments onto the barges deck, and considering the effects of this movement＇s speed and variations in tidal levels during the move. Then the model was solved by the penalty function method, the grid method, and improved simulated annealing, respectively. The best optimization model and its corresponding solution were then determined. Finally, it was proven that the model and the method adopted are correct and suitable, by calculating and analysing an example.

An Improved Control Vector Iteration Approach for Nonlinear Dynamic Optimization. II. Problems with Path Constraints

This paper considers dealing with path constraints in the framework of the improved control vector iteration （CVI） approach. Two available ways for enforcing equality path constraints are presented, which can be directly incorporated into the improved CVI approach. Inequality path constraints are much more difficult to deal with, even for small scale problems, because the time intervals where the inequality path constraints are active are unknown in advance. To overcome the challenge, the ll penalty function and a novel smoothing technique are in-troduced, leading to a new effective approach. Moreover, on the basis of the relevant theorems, a numerical algo-rithm is proposed for nonlinear dynamic optimization problems with inequality path constraints. Results obtained from the classic batch reaCtor operation problem are in agreement with the literature reoorts, and the comoutational efficiency is also high.

Exact Penalty Method for the Nonlinear Bilevel Programming Problem

In this paper,following the method of replacing the lower level problem with its Kuhn-Tucker optimality condition,we transform the nonlinear bilevel programming problem into a normal nonlinear programming problem with the complementary slackness constraint condition.Then,we get the penalized problem of the normal nonlinear programming problem by appending the complementary slackness condition to the upper level objective with a penalty.We prove that this penalty function is exact and the penalized problem and the nonlinear bilevel programming problem have the same global optimal solution set.Finally,we propose an algorithm for the nonlinear bilevel programming problem.The numerical results show that the algorithm is feasible and efficient.

Optimization Mathematical Model of Pile Forces for Offshore Piled Breasting Dolphins

An optimization mathematical model of the pile forces for piled breasting dolphins in the open sea under various loading conditions is presented. The optimum layout with the well distributed pile forces and the least number of piles is achieved by the multiplier penalty function method. Several engineering cases have been calculated and compared with the result of the conventional design method. It is shown that the number of piles can be reduced at least by 10%～20% and the piles' bearing state is improved greatly.

New Approach to Linear Stackelberg Problems with Multiple Leaders-followers

In this paper, we study linear static Stac kelberg problems with multiple leaders-followers in which each decision maker wi thin his group may or may not cooperate. An exact penalty function method is dev eloped. The duality gaps of the followers’ problems are appended to the leaders’ objective function with a penalty. The structure leads to the decomposition of the composite problem into a series of linear programmings leading to an efficie nt algorithm. We prove that local optimality is reached for an exact penalty fun ction and illustrate the method with three examples. The model in this paper ext ends the stackelberg leader-follower model.

THREE-DIMENSIONAL NONLINEAR ANALYSIS OF DRILL STRING STRUCTURE IN ANNUI US

Three-dimensional nonlinear analysis of drill string structure in annulus of curvedwellbore is done by using the theory of finite element and Newton-Raphson method.According to the characteristics of its deformation,a method of the description andcomputation of taking different forms of elements for different parameters is advanced.The penalty function method is applied for finding the unknown boundary .the nonlinear effects of curvature of wellbore on the side forces on bit ae shown by thecomputation.

BILEVEL PROGRAMMING MODEL AND SOLUTION METHOD FOR MIXED TRANSPORTATION NETWORK DESIGN PROBLEM

By handling the travel cost function artfully, the authors formulate the transportation mixed network design problem （MNDP） as a mixed-integer, nonlinear bilevel programming problem, in which the lower-level problem, comparing with that of conventional bilevel DNDP models, is not a side constrained user equilibrium assignment problem, but a standard user equilibrium assignment problem. Then, the bilevel programming model for MNDP is reformulated as a continuous version of bilevel programming problem by the continuation method. By virtue of the optimal-value function, the lower-level assignment problem can be expressed as a nonlinear equality constraint. Therefore, the bilevel programming model for MNDP can be transformed into an equivalent single-level optimization problem. By exploring the inherent nature of the MNDP, the optimal-value function for the lower- level equilibrium assignment problem is proved to be continuously differentiable and its functional value and gradient can be obtained efficiently. Thus, a continuously differentiable but still nonconvex optimization formulation of the MNDP is created, and then a locally convergent algorithm is proposed by applying penalty function method. The inner loop of solving the subproblem is mainly to implement an Ml-or-nothing assignment. Finally, a small-scale transportation network and a large-scale network are presented to verify the proposed model and algorithm.

A Continuation Algorithm for Max-Cut Problem

A continuation algorithm for the solution of max-cut problems is proposed in this paper. Unlike the available semi-definite relaxation, a max-cut problem is converted into a continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved by using the augmented Lagrange penalty function method. The convergence property of the proposed algorithm is studied. Numerical experiments and comparisons with the Geomeans and Williamson randomized algorithm made on some max-cut test problems show that the algorithm generates satisfactory solutions for all the test problems with much less computation costs.

MATHEMATICAL ANALYSIS OF SOME NEURAL NETWORKS FOR SOLVING LINEAR AND QUADRATIC PROGRAMMING

Artificial neural network techniques have been introduced into the area of optimization in the recent decade. Some neural network models have been suggested to solve linear and quadratic programming problems. The Kennedy and Chua model[5] is one of these networks. In this paper results about the convergence of the model are obtained. Another related problem is how to choose a parameter value s so that the equilibrium point of the network immediately and properly approximates the original solution. Such an estimation for the parameter is given in a closed form when the network is used to solve linear programming.