Primal-Dual Interior-Point Algorithms with Dynamic Step-Size Based on Kernel Functions for Linear Programming

In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming （LP） problems. The algorithms are based on a few kernel functions, including both serf-regular functions and non-serf-regular ones. The dynamic step size is compared with fixed step size for the algorithms in inner iteration of Newton step. Numerical tests show that the algorithms with dynaraic step size are more efficient than those with fixed step size.

A Primal-Dual Infeasible-Interior-Point Algorithm for Multiple Objective Linear Programming Problems

A primal-dual infeasible interior point algorithm for multiple objective linear programming (MOLP) problems was presented. In contrast to the current MOLP algorithm. moving through the interior of polytope but not confining the iterates within the feasible region in our proposed algorithm result in a solution approach that is quite different and less sensitive to problem size, so providing the potential to dramatically improve the practical computation effectiveness.

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.

A NEW FRAMEWORK OF PRIMAL-DUAL INFEASIBLE INTERIOR-POINT METHOD FOR LINEAR PROGRAMMING

On the basis of the formulations of the logarithmic barrier function and the idea of following the path of minimizers for the logarithmic barrier family of problems the so called "centralpath" for linear programming, we propose a new framework of primal-dual infeasible interiorpoint method for linear programming problems. Without the strict convexity of the logarithmic barrier function, we get the following results: (a) if the homotopy parameterμcan not reach to zero,then the feasible set of these programming problems is empty; (b) if the strictly feasible set is nonempty and the solution set is bounded, then for any initial point x, we can obtain a solution of the problems by this method; (c) if the strictly feasible set is nonempty and the solution set is unbounded, then for any initial point x, we can obtain a (?)-solution; and(d) if the strictly feasible set is nonempty and the solution set is empty, then we can get the curve x(μ), which towards to the generalized solutions.

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.

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.

A POLYNOMIAL PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O（√nL） iteration complexity which is the best result for convex quadratic programming so far.

An Improved Affine-Scaling Interior Point Algorithm for Linear Programming

In this paper, an Improved Affine-Scaling Interior Point Algorithm for Linear Programming has been proposed. Computational results of selected practical problems affirming the proposed algorithm have been provided. The proposed algorithm is accurate, faster and therefore reduces the number of iterations required to obtain an optimal solution of a given Linear Programming problem as compared to the already existing Affine-Scaling Interior Point Algorithm. The algorithm can be very useful for development of faster software packages for solving linear programming problems using the interior-point methods.

Predictor-corrector interior-point algorithm for linearly constrained convex programming

Active set method and gradient projection method are curre nt ly the main approaches for linearly constrained convex programming. Interior-po int method is one of the most effective choices for linear programming. In the p aper a predictor-corrector interior-point algorithm for linearly constrained c onvex programming under the predictor-corrector motivation was proposed. In eac h iteration, the algorithm first performs a predictor-step to reduce the dualit y gap and then a corrector-step to keep the points close to the central traject ory. Computations in the algorithm only require that the initial iterate be nonn egative while feasibility or strict feasibility is not required. It is proved th at the algorithm is equivalent to a level-1 perturbed composite Newton method. Numerical experiments on twenty-six standard test problems are made. The result s show that the proposed algorithm is stable and robust.

A PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.

AN INFEASIBLE-INTERIOR-POINT PREDICTOR-CORRECTOR ALGORITHM FOR THE SECOND-ORDER CONE PROGRAM

A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.

Linear Programming for Optimum PID Controller Tuning

This work presents a new methodology based on Linear Programming (LP) to tune Proportional-Integral-Derivative (PID) control parameters. From a specification of a desired output time domain of the plant, a linear optimization system is proposed to adjust the PID controller leading the output signal to stable operation condition with minimum oscillations. The constraint set used in the optimization process is defined by using numerical integration approach. The generated optimization problem is convex and easily solved using an interior point algorithm. Results obtained using familiar plants from literature have shown that the proposed linear programming problem is very effective for tuning PID controllers.

A New Heuristic for the Convex Quadratic Programming Problem

This paper presents a new heuristic to linearise the convex quadratic programming problem. The usual Karush-Kuhn-Tucker conditions are used but in this case a linear objective function is also formulated from the set of linear equations and complementarity slackness conditions. An unboundedness challenge arises in the proposed formulation and this challenge is alleviated by construction of an additional constraint. The formulated linear programming problem can be solved efficiently by the available simplex or interior point algorithms. There is no restricted base entry in this new formulation. Some computational experiments were carried out and results are provided.

A Primal-dual Interior-point Algorithm for Symmetric Cone Convex Quadratic Programming Based on the Commutative Class Directions

In this paper, we present a neighborhood following primal-dual interior-point algorithm for solving symmetric cone convex quadratic programming problems, where the objective function is a convex quadratic function and the feasible set is the intersection of an affine subspace and a symmetric cone attached to a Euclidean Jordan algebra. The algorithm is based on the [13] broad class of commutative search directions for cone of semidefinite matrices, extended by [18] to arbitrary symmetric cones. Despite the fact that the neighborhood is wider, which allows the iterates move towards optimality with longer steps, the complexity iteration bound remains as the same result of Schmieta and Alizadeh for symmetric cone optimization problems.

Two new predictor-corrector algorithms for second-order cone programming

Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming （SOCP） are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton （AHO） directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O（rln（εo/ε）） is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.

Optimal Adjustment Algorithm for <i>p</i>Coordinates and The Starting Point in Interior Point Methods

Optimal adjustment algorithm for p coordinates is a generalization of the optimal pair adjustment algorithm for linear programming, which in turn is based on von Neumann’s algorithm. Its main advantages are simplicity and quick progress in the early iterations. In this work, to accelerate the convergence of the interior point method, few iterations of this generalized algorithm are applied to the Mehrotra’s heuristic, which determines the starting point for the interior point method in the PCx software. Computational experiments in a set of linear programming problems have shown that this approach reduces the total number of iterations and the running time for many of them, including large-scale ones.

A NEW DETERMINISTIC FORMULATION FOR DYNAMIC STOCHASTIC PROGRAMMING PROBLEMS AND ITS NUMERICAL COMPARISON WITH OTHERS

A new deterministic formulation,called the conditional expectation formulation,is proposed for dynamic stochastic programming problems in order to overcome some disadvantages of existing deterministic formulations.We then check the impact of the new deterministic formulation and other two deterministic formulations on the corresponding problem size,nonzero elements and solution time by solving some typical dynamic stochastic programming problems with different interior point algorithms.Numerical results show the advantage and application of the new deterministic formulation.

A Modified Full-NT-Step Infeasible Interior-Point Algorithm for SDP Based on a Specific Kernel Function

This paper proposes a new full Nesterov-Todd(NT) step infeasible interior-point algorithm for semidefinite programming. Our algorithm uses a specific kernel function, which is adopted by Liu and Sun, to deduce the feasibility step. By using the step, it is remarkable that in each iteration of the algorithm it needs only one full-NT step, and can obtain an iterate approximate to the central path. Moreover, it is proved that the iterative bound corresponds with the known optimal one for semidefinite optimization problems.

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.