New approach for the nonlinear programming with transient stability constraints arising from power systems

This paper presents a new approach for solving a class of complicated nonlinear programming problems arises from optimal power flow with transient stability constraints (denoted by OTS) in power systems. By using a functional transformation technology proposed in Chen et al. (IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48:327–339, [2001]), the OTS problem is transformed to a semi-infinite programming (SIP). Then based on the KKT (Karush-Kuhn-Tucker) system of the reformulated SIP problem and the finite approximation technology, an iterative method is presented, which develops Wu-Li-Qi-Zhou’ (Optim. Methods Softw. 20:629–643, [2005]) method. In order to save the computing cost, some typical computing technologies, such as active set strategy, quasi-Newton method for the subproblems coming from the finite approximation model, are addressed. The global convergence of the proposed algorithm is established. Numerical examples from power systems are tested. The computing results show the efficiency of the new approach.     

An optimal method for stochastic composite optimization

This paper considers an important class of convex programming (CP) problems, namely, the stochastic composite optimization (SCO), whose objective function is given by the summation of general nonsmooth and smooth stochastic components. Since SCO covers non-smooth, smooth and stochastic CP as certain special cases, a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming. Note however that the optimization algorithms that can achieve this lower bound had never been developed. In this paper, we show that the simple mirror-descent stochastic approximation method exhibits the best-known rate of convergence for solving these problems. Our major contribution is to introduce the accelerated stochastic approximation (AC-SA) algorithm based on Nesterov’s optimal method for smooth CP (Nesterov in Doklady AN SSSR 269:543–547, 1983; Nesterov in Math Program 103:127–152, 2005), and show that the AC-SA algorithm can achieve the aforementioned lower bound on the rate of convergence for SCO. To the best of our knowledge, it is also the first universally optimal algorithm in the literature for solving non-smooth, smooth and stochastic CP problems. We illustrate the significant advantages of the AC-SA algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems.     

Implementing an efficient fptas for the 0–1 multi-objective knapsack problem

In the present work, we are interested in the practical behavior of a new fully polynomial time approximation schemes (fptas) to solve the approximation version of the 0–1 multi-objective knapsack problem. The proposed methodology makes use of very general techniques (such as dominance relations in dynamic programming) and thus may be applicable in the implementation of fptas for other problems as well.     

Linear programming in measure spaces

This paper studies a linear programming problem in measure spaces (LPM). Several results are obtained. First, the optimal value of LPM can be equal to the optimal value of the dual problem (DLPM), but the solution of DLPM may be not exist in its feasible region. Sccond, :he relations between the optimal solution of LPM and the extreme point of the feasible region of LPM are discussed. In order to investigate the conditions under which a feasible solution becomes an extremal point, the inequality constraint of LPM is transformed to an equality constraint. Third, the LPM can be reformulated to be a general capacity problem (GCAP) or a linear semi-infinite programming problem (LSIP = SIP), and under appropriate restrictioiis, the algorithm developed by the authors in [7] and [8] are applicable for developing an approximation scheme for the optimal solution of LPM     

Convex Semi-Infinite Programming: Implicit Optimality Criterion Based on the Concept of Immobile Indices

We state a new implicit optimality criterion for convex semi-infinite programming (SIP) problems. This criterion does not require any constraint qualification and is based on concepts of immobile index and immobility order. Given a convex SIP problem with a continuum of constraints, we use an information about its immobile indices to construct a nonlinear programming (NLP) problem of a special form. We prove that a feasible point of the original infinite SIP problem is optimal if and only if it is optimal in the corresponding finite NLP problem. This fact allows us to obtain new efficient optimality conditions for convex SIP problems using known results of the optimality theory of NLP. To construct the NLP problem, we use the DIO algorithm. A comparison of the optimality conditions obtained in the paper with known results is provided.     

Optimal rates of linear convergence of the averaged alternating modified reflections method for two subspaces

Numerical Algorithms - The main difficulty for solving semi-infinite programming (SIP) problem is precisely that it has infinitely many constraints. By using a maximum function, the SIP problem can...     

A sequential quadratic programming with a dual parametrization approach to nonlinear semi-infinite programming

In the sequel of the work reported in Liu et al. (1999), in which a method based on a dual parametrization is used to solve linear-quadratic semi-infinite programming (SIP) problems, a sequential quadratic programming technique is proposed to solve nonlinear SIP problems. A merit function to measure progress toward the solution and a procedure to compute the penalty parameter are also proposed.     

Outer approximation schemes for generalized semi-infinite variational inequality problems

We introduce and analyse outer approximation schemes for solving variational inequality problems in which the constraint set is as in generalized semi-infinite programming. We call these problems generalized semi-infinite variational inequality problems. First, we establish convergence results of our method under standard boundedness assumptions. Second, we use suitable Tikhonov-like regularizations for establishing convergence in the unbounded case.     

Integrating operations research in constraint programming

This paper presents constraint programming (CP) as a natural formalism for modelling problems, and as a flexible platform for solving them. CP has a range of techniques for handling constraints including several forms of propagation and tailored algorithms for global constraints. It also allows linear programming to be combined with propagation and novel and varied search techniques which can be easily expressed in CP. The paper describes how CP can be used to exploit linear programming within different kinds of hybrid algorithm. In particular it can enhance techniques such as Lagrangian relaxation, Benders decomposition and column generation.     

Global solution of semi-infinite programs

Optimization problems involving a finite number of decision variables and an infinite number of constraints are referred to as semi-infinite programs (SIPs). Existing numerical methods for solving nonlinear SIPs make strong assumptions on the properties of the feasible set, e.g., convexity and/or regularity, or solve a discretized approximation which only guarantees a lower bound to the true solution value of the SIP. Here, a general, deterministic algorithm for solving semi-infinite programs to guaranteed global optimality is presented. A branch-and-bound (B&B) framework is used to generate convergent sequences of upper and lower bounds on the SIP solution value. The upper-bounding problem is generated by replacing the infinite number of real-valued constraints with a finite number of constrained inclusion bounds; the lower-bounding problem is formulated as a convex relaxation of a discretized approximation to the SIP. The SIP B&B algorithm is shown to converge finitely to –optimality when the subdivision and discretization procedures used to formulate the node subproblems are known to retain certain convergence characteristics. Other than the properties assumed by globally-convergent B&B methods (for finitely-constrained NLPs), this SIP algorithm makes only one additional assumption: For every minimizer x* of the SIP there exists a sequence of Slater points x_n for which (cf. Section 5.4). Numerical results for test problems in the SIP literature are presented. The exclusion test and a modified upper-bounding problem are also investigated as heuristic approaches for alleviating the computational cost of solving a nonlinear SIP to guaranteed global optimality.     

Optimality conditions for non-smooth semi-infinite programming

This article deals with a class of non-smooth semi-infinite programming (SIP) problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. We introduce several kinds of constraint qualifications for these non-smooth SIP problems and we study the relationships between them. Finally, necessary and sufficient optimality conditions are investigated.     

Study of a special nonlinear problem arising in convex semi-infinite programming

We consider convex problems of semi-infinite programming (SIP) using an approach based on the implicit optimality criterion. This criterion allows one to replace optimality conditions for a feasible solution x ⁰ of the convex SIP problem by such conditions for x ⁰ in some nonlinear programming (NLP) problem denoted by NLP(I(x ⁰)). This nonlinear problem, constructed on the base of special characteristics of the original SIP problem, so-called immobile indices and their immobility orders, has a special structure and a diversity of important properties. We study these properties and use them to obtain efficient explicit optimality conditions for the problem NLP(I(x ⁰)). Application of these conditions, together with the implicit optimality criterion, gives new efficient optimality conditions for convex SIP problems. Special attention is paid to SIP problems whose constraints do not satisfy the Slater condition and to problems with analytic constraint functions for which we obtain optimality conditions in the form of a criterion. Comparison with some known optimality conditions for convex SIP is provided.     

Solving semi-infinite programs by smoothing projected gradient method

In this paper, we study a semi-infinite programming (SIP) problem with a convex set constraint. Using the value function of the lower level problem, we reformulate SIP problem as a nonsmooth optimization problem. Using the theory of nonsmooth Lagrange multiplier rules and Danskin’s theorem, we present constraint qualifications and necessary optimality conditions. We propose a new numerical method for solving the problem. The novelty of our numerical method is to use the integral entropy function to approximate the value function and then solve SIP by the smoothing projected gradient method. Moreover we study the relationships between the approximating problems and the original SIP problem. We derive error bounds between the integral entropy function and the value function, and between locally optimal solutions of the smoothing problem and those for the original problem. Using certain second order sufficient conditions, we derive some estimates for locally optimal solutions of problem. Numerical experiments show that the algorithm is efficient for solving SIP.     

A New Quadratic Semi-infinite Programming Algorithm Based on Dual Parametrization

The so called dual parameterization method for quadratic semi-infinite programming (SIP) problems is developed recently. A dual parameterization algorithm is also proposed for numerical solution of such problems. In this paper, we present and improved adaptive algorithm for quadratic SIP problems with positive definite objective and multiple linear infinite constraints. In each iteration of the new algorithm, only a quadratic programming problem with a limited dimension and a limited number of constraints is required to be solved. Furthermore, convergence result is given. The efficiency of the new algorithm is shown by solving a number of numerical examples.     

0-1多项式规划问题的SDP松弛方法(英)

本文提出了一类新的构造0-1多项式规划的半定规划(SDP)松弛方法. 我们首先利用矩阵分解和分片线性逼近给出一种新的SDP松弛, 该 松弛产生的界比标准线性松弛产生的界更紧. 我们还利用 拉格朗日松弛和平方和(SOS)松弛方法给出了一种构造Lasserre的SDP 松弛的新方法.     

Approximations of Relaxed Optimal Control Problems

In the present paper, we investigate an approximation technique for relaxed optimal control problems. We study control processes governed by ordinary differential equations in the presence of state, target, and integral constraints. A variety of approximation schemes have been recognized as powerful tools for the theoretical studying and practical solving of Infinite-dimensional optimization problems. On the other hand, theoretical approaches to the relaxed optimal control problem with constraints are not sufficiently advanced to yield numerically tractable schemes. The explicit approximation of the compact control set makes it possible to reduce the sophisticated relaxed problem to an auxiliary optimization problem. A given trajectory of the relaxed problem can be approximated by trajectories of the auxiliary problem. An optimal solution of the introduced optimization problem provides a basis for the construction of minimizing sequences for the original optimal control problem. We describe how to carry out the numerical calculations in the context of nonlinear programming and establish the convergence properties of the obtained approximations.The authors thank the referees for helpful comments and suggestions.     

An Adaptive Dual Parametrization Algorithm for Quadratic Semi-infinite Programming Problems

The so called dual parametrization method for quadratic semi-infinite programming (SIP) problems is developed recently for quadratic SIP problems with a single infinite constraint. A dual parametrization algorithm is also proposed for numerical solution of such problems. In this paper, we consider quadratic SIP problems with positive definite objective and multiple linear infinite constraints. All the infinite constraints are supposed to be continuously dependent on their index variable on a compact set which is defined by a number equality and inequalities. We prove that in the multiple infinite constraint case, the minimu parametrization number, just as in the single infinite constraint case, is less or equal to the dimension of the SIP problem. Furthermore, we propose an adaptive dual parametrization algorithm with convergence result. Compared with the previous dual parametrization algorithm, the adaptive algorithm solves subproblems with much smaller number of constraints. The efficiency of the new algorithm is shown by solving a number of numerical examples.     

Duality and Correction of Inconsistent Constraints for Improper Linear Programming Problems

We continue the study of approximation properties of alternative duality schemes for improper problems of linear programming. The schemes are based on the use of the classical Lagrange function regularized simultaneously in primal and dual variables. The earlier results on the connection of its saddle points with the lexicographic correction of the right-hand sides of constraints in improper problems of the first and second kind are transferred to a more general type of improperness. Convergence theorems are presented and an informal interpretation of the obtained generalized solution is given.     

A new smoothing Newton-type algorithm for semi-infinite programming

We consider a semismooth reformulation of the KKT system arising from the semi-infinite programming (SIP) problem. Based upon this reformulation, we present a new smoothing Newton-type method for the solution of SIP problem. The main properties of this method are: (a) it is globally convergent at least to a stationary point of the SIP problem, (b) it is locally superlinearly convergent under a certain regularity condition, (c) the feasibility is ensured via the aggregated constraint, and (d) it has to solve just one linear system of equations at each iteration. Preliminary numerical results are reported.     

Quadratic approximation of penalty functions for solving large-scale linear programs

Using the least squares, modified Lagrangian function, and some other methods as examples, the capabilities of the new optimization technique based on the quadratic approximation of penalty functions that has been recently proposed by O. Mangasarian for a special class of linear programming problems are demonstrated. The application of this technique makes it possible to use unified matrix operations and standard linear algebra packages (including parallel ones) for solving large-scale problems with sparse strongly structured constraint matrices. With this technique, the computational schemes of some well-known algorithms can take an unexpected form.