A Note on the Existence of a Specified Number of Interior Points

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let h（κ） be the smallest integer such that every set of points in the plane, no three collinear, with at least h（κ） interior points, has a subset of points with exactly κ or κ ＋ 1 interior points of P. We prove that h（5）=11.

Several Inequalities Involving Two Simplexes and Interior Points

This article establishes several new geometric inequalities, which refer to the lengthes of the edges of a simplex and interior point, height, lateral area, and the circumradius of another simplex.

A Combined Homotopy Infeasible Interior-Point Method for Convex Nonlinear Programming

In this paper, on the basis of the logarithmic barrier function and KKT conditions, we propose a combined homotopy infeasible interior-point method （CHIIP） for convex nonlinear programming problems. For any convex nonlinear programming, without strict convexity for the logarithmic barrier function, we get different solutions of the convex programming in different cases by CHIIP method.

A predictor-corrector interior-point algorithmfor monotone variational inequality problems

Mehrotra's recent suggestion of a predictor corrector variant of primal dual interior point method for linear programming is currently the interior point method of choice for linear programming. In this work the authors give a predictor corrector interior point algorithm for monotone variational inequality problems. The algorithm was proved to be equivalent to a level 1 perturbed composite Newton method. Computations in the algorithm do not require the initial iteration to be feasible. Numerical results of experiments are presented.

Dynamic economic dispatch combining network flow and interior point method

Under the environment of electric power market, economic dispatch (ED) problem should consider network constraints, unit ramp rates, besides the basic constraints. For this problem, it is important to establish the effective model and algorithm. This paper examines the decoupled conditions that affect the solution optimality to this problem. It proposes an effective model and solution method. Based on the look-ahead technique, it finds the number of time intervals to guarantee the solution optimality. Next, an efficient technique for finding the optimal solution via the interior point methods is described. Test cases, which include dispatching six units over 5 time intervals on the IEEE 30 test system with line flows and ramp constraints are presented. Results indicate that the computational effort as measured by iteration counts or execution time varies only modestly with the problem size.

Penalized interior point approach for constrained nonlinear programming

A penalized interior point approach for constrained nonlinear programming is examined in this work. To overcome the difficulty of initialization for the interior point method, a problem equivalent to the primal problem via incorporating an auxiliary variable is constructed. A combined approach of logarithm barrier and quadratic penalty function is proposed to solve the problem. Based on Newton＇s method, the global convergence of interior point and line search algorithm is proven. Only a finite number of iterations is required to reach an approximate optimal solution. Numerical tests are given to show the effectiveness of the method.

Element-free Galerkin method for free vibration of rectangular plates with interior elastic point supports and elastically restrained edges

The element-free Galerkin method is proposed to solve free vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges.Based on the extended Hamilton＇s principle for the elastic dynamics system,the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established using the element-free Galerkin method.Through numerical calculation,curves of the natural frequency of thin plates with three edges simply supported and one edge elastically restrained,and three edges clamped and the other edge elastically restrained versus the spring constant,locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained.Effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.

A Primal-dual Interior Point Method for Nonlinear Programming

In this paper, we propose a primal-dual interior point method for solving general constrained nonlinear programming problems. To avoid the situation that the algorithm we use may converge to a saddle point or a local maximum, we utilize a merit function to guide the iterates toward a local minimum. Especially, we add the parameter ε to the Newton system when calculating the decrease directions. The global convergence is achieved by the decrease of a merit function. Furthermore, the numerical results confirm that the algorithm can solve this kind of problems in an efficient way.

Solving Fixed Point Problems in More General Nonconvex Sets Via an Interior Point Homotopy Method

In this paper,we are mainly devoted to solving fixed point problems in more general nonconvex sets via an interior point homotopy method.Under suitable conditions,a constructive proof is given to prove the existence of fixed points,which can lead to an implementable globally convergent algorithm.

Globally Convergent Interior Point Methods for Variational Inequalities in Unbounded Sets

The finite-dimensional variational inequality problem (VIP) has been studied extensively in the literature because of its successful applications in many fields such as economics, transportation, regional science and operations research. Barker and Pang[1] have given an excellent survey of theories, methods and applications of VIPs.

Power Generation Expansion Planning Using an Interior Point with Cutting Plane （IP/CP） Method

The generation expansion planning is one of complex mixed-integer optimization problems, which involves a large number of continuous or discrete decision variables and constraints. In this paper, an interior point with cutting plane （IP/CP） method is proposed to solve the mixed-integer optimization problem of the electrical power generation expansion planning. The IP/CP method could improve the overall efficiency of the solution and reduce the computational time. Proposed method is combined with the Bender＇s decomposition technique in order to decompose the generation expansion problem into a master investment problem and a slave operational problem. The numerical example is presented to compare with the effectiveness of the proposed algorithm.

Low-order Wavefront Error Compensation for Multi-field of Lithography Projection Objective Based on Interior Point Method

Low-order wavefront error account for a large proportion of wave aberrations.A compensation method for low order aberration of projection lithography objective based on Interior Point Method is presented.Compensation model between wavefront error and degree of movable lens freedom is established.Converting over-determined system to underdetermined system,the compensation is solved by Interior Point Method(IPM).The presented method is compared with direct solve the over-determined system.Then,other algorithm GA,EA and PS is compared with IPM.Simulation and experimental results show that the presented compensation method can obtained compensation with less residuals compared with direct solve the over-determined system.Also,the presented compensation method can reduce computation time and obtain results with less residuals compare with AGA,EA and PS.Moreover,after compensation,RMS of wavefront error of the experimental lithography projection objective decrease from 56.05 nm to 17.88 nm.

Joint Beam Scheduling and Power Optimization for Beam Hopping LEO Satellite Systems

Low earth orbit(LEO) satellite communications can provide ubiquitous and reliable services,making it an essential part of the Internet of Everything network. Beam hopping(BH) is an emerging technology for effectively addressing the issue of low resource utilization caused by the non-uniform spatio-temporal distribution of traffic demands. However, how to allocate multi-dimensional resources in a timely and efficient way for the highly dynamic LEO satellite systems remains a challenge. This paper proposes a joint beam scheduling and power optimization beam hopping(JBSPO-BH) algorithm considering the differences in the geographic distribution of sink nodes. The JBSPO-BH algorithm decouples the original problem into two sub-problems. The beam scheduling problem is modelled as a potential game,and the Nash equilibrium(NE) point is obtained as the beam scheduling strategy. Moreover, the penalty function interior point method is applied to optimize the power allocation. Simulation results show that the JBSPO-BH algorithm has low time complexity and fast convergence and achieves better performance both in throughput and fairness. Compared with greedybased BH, greedy-based BH with the power optimization, round-robin BH, Max-SINR BH and satellite resource allocation algorithm, the throughput of the proposed algorithm is improved by 44.99%, 20.79%,156.06%, 15.39% and 8.17%, respectively.

A novel DPSS filter optimization scheme to reduce the intrinsic interference of FBMC-QAM systems

In order to reduce the intrinsic interference of the filter bank multicarrier-quadrature amplitude modulation(FBMC-QAM)system,a novel filter optimization scheme based on discrete prolate spheroidal sequences(DPSS)is proposed.Firstly,a prototype filter function based on DPSS is designed,since the eigenvalue can be used as an indicator of the energy concentration of DPSS,so a threshold is set,and the sequence with the most concentrated energy is selected under the threshold,that is,the sequence with the eigenvalue higher than the threshold,and the prototype filter function is rewritten as a weighted sum function of multiple eigenvectors.Under the energy constraints of the filter,the relationship between the eigenvectors and the intrinsic interference function is established,and the function problem is transformed into an optimization problem for the weighted coefficients.Through the interior point method,the most suitable weight is found to obtain the minimum intrinsic interference result.Theoretical analysis and simulation results show that compared with the prototype filters such as Type1 and CaseC,the DPSS filter applying the proposed optimization algorithm can effectively suppress the intrinsic interference of the system and obtain a better bit error rate(BER)performance.

Nash Equilibrium of a Fixed-Sum Two-Player Game

It is well established that Nash equilibrium exists within the framework of mixed strategies in strategic-form non-cooperative games. However, finding the Nash equilibrium generally belongs to the class of problems known as PPAD (Polynomial Parity Argument on Directed graphs), for which no polynomial-time solution methods are known, even for two-player games. This paper demonstrates that in fixed-sum two-player games (including zero-sum games), the Nash equilibrium forms a convex set, and has a unique expected payoff. Furthermore, these equilibria are Pareto optimal. Additionally, it is shown that the Nash equilibrium of fixed-sum two-player games can theoretically be found in polynomial time using the principal-dual interior point method, a solution method of linear programming.

Numerical simulation of a gas pipeline network using computational fluid dynamics simulators

This article describes numerical simulation of gas pipeline network operation using high-accuracy computational fluid dynamics (CFD) simulators of the modes of gas mixture transmission through long, multi-line pipeline systems (CFD-simulator). The approach used in CFD-simulators for modeling gas mixture transmission through long, branched, multi-section pipelines is based on tailoring the full system of fluid dynamics equations to conditions of unsteady, non-isothermal processes of the gas mixture flow. Identification, in a CFD-simulator, of safe parameters for gas transmission through compressor stations amounts to finding the interior points of admissible sets described by systems of nonlinear algebraic equalities and inequalities. Such systems of equalities and inequalities comprise a formal statement of technological, design, operational and other constraints to which operation of the network equipment is subject. To illustrate the practicability of the method of numerical simulation of a gas transmission network, we compare computation results and gas flow parameters measured on-site at the gas transmission enter-prise.

A novel PID controller tuning method based on optimization technique

An approach for parameter estimation of proportional-integral-derivative(PID) control system using a new nonlinear programming(NLP) algorithm was proposed.SQP/IIPM algorithm is a sequential quadratic programming(SQP) based algorithm that derives its search directions by solving quadratic programming(QP) subproblems via an infeasible interior point method(IIPM) and evaluates step length adaptively via a simple line search and/or a quadratic search algorithm depending on the termination of the IIPM solver.The task of tuning PI/PID parameters for the first-and second-order systems was modeled as constrained NLP problem. SQP/IIPM algorithm was applied to determining the optimum parameters for the PI/PID control systems.To assess the performance of the proposed method,a Matlab simulation of PID controller tuning was conducted to compare the proposed SQP/IIPM algorithm with the gain and phase margin(GPM) method and Ziegler-Nichols(ZN) method.The results reveal that,for both step and impulse response tests,the PI/PID controller using SQP/IIPM optimization algorithm consistently reduce rise time,settling-time and remarkably lower overshoot compared to GPM and ZN methods,and the proposed method improves the robustness and effectiveness of numerical optimization of PID control systems.

Risk Control of Transmission Line Overload for Wind-Integrated Power Systems

With the rapid development of the wind generation,uncertainties of random wind and load bring some inevitable impacts on the security of power system. Once the uncertainty causes line power to exceed its limit, line overload will occur. The paper presents the risk control of transmission line overload for windintegrated power systems. Firstly, a risk control model of line overload is proposed considering the uncertainties of loads,generator outputs and wind powers. The generation cost and security level of system associated with overload can be optimally controlled. Then path following interior point method is employed to carry out the optimal control. Finally the simulation is made on the modified IEEE-30 bus system. Results show that the risk of line overload is effectively reduced through the optimization of control variables.