Algorithm For Semi-infinite Programming Research Articles

An inexact primal-dual algorithm for semi-infinite programming

This paper considers an inexact primal-dual algorithm for semi-infinite programming (SIP) for which it provides general error bounds. To implement the dual variable update, we create a new prox function for nonnegative measures which turns out to be a generalization of the Kullback-Leibler divergence for probability distributions. We show that under suitable conditions on the error, this algorithm achieves an $\mathcal{O}(1/\sqrt{K})$ rate of convergence in terms of the optimality gap and constraint violation. We then use our general error bounds to analyze the convergence and sample complexity of a specific primal-dual SIP algorithm based on Monte Carlo integration. Finally, we provide numerical experiments to demonstrate the performance of our algorithm.

A smoothing Levenberg–Marquardt algorithm for semi-infinite programming

In this paper, we present a smoothing Levenberg---Marquardt algorithm for the solution of the semi-infinite programming (SIP) problem. We first reformulate the KKT system of SIP problem into a system of constrained nonsmooth equations. Then we solve this system by a smoothing Levenberg---Marquardt algorithm. The feasibility is ensured via the aggregated constraint, and at each iteration of the presented algorithm only a quadratic programming has to be solved. Global and local superlinear convergence of this algorithm is established under a local error bound condition, which is much weaker than the nonsingularity condition. Preliminary numerical results are reported.

On the convergence of a smoothed penalty algorithm for semi-infinite programming

For semi-infinite programming (SIP), we consider a class of smoothed penalty functions, which approximate the exact $$l_\rho (0<\rho \le 1)$$ penalty functions. On base of the smoothed penalty function, we present a feasible penalty algorithm for solving SIP. Without any boundedness condition or coercive condition, we establish the global convergence theorem. Then we present a perturbation theorem for this algorithm and obtain a necessary and sufficient condition for the convergence to the optimal value of SIP. Under Mangasarian–Fromovitz constrained qualification condition, we further discuss the convergence properties for the algorithm based upon a subclass of smooth approximations to the exact $$l_\rho $$ penalty function. Finally, numerical results are given.

A semi-infinite programming approach to identifying matrix-exponential distributions

The Laplace–Stieltjes transform of a matrix-exponential (ME) distribution is a rational function where at least one of its poles of maximal real part is real and negative. The coefficients of the numerator polynomial, however, are more difficult to characterise. It is known that they are contained in a bounded convex set that is the intersection of an uncountably infinite number of linear half-spaces. In order to determine whether a given vector of numerator coefficients is contained in this set (i.e. the vector corresponds to an ME distribution) we present a semi-infinite programming algorithm that minimises a convex distance function over the set. In addition, in the event that the given vector does not correspond to an ME distribution, the algorithm returns a closest vector which does correspond to one.

The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets

A numerical solution method for semi-infinite optimization problems with arbitrary, not necessarily box-shaped, index sets is presented. Following the ideas of Floudas and Stein (SIAM J Optim 18:1187–1208, 2007), convex relaxations of the lower level problem are adaptively constructed and then reformulated as mathematical programs with complementarity constraints and solved. Although the index set is arbitrary, this approximation produces feasible iterates for the original problem. The convex relaxations and needed parameters are constructed with ideas of the αBB method of global optimization and interval methods. It is shown that after finitely many steps an $${\epsilon}$$ -stationary point of the original semi-infinite problem is reached. A numerical example illustrates the performance of the proposed method.

An Augmented Lagrangian Algorithm for Solving SemiinfiniteProgramming

We present a smooth augmented Lagrangian algorithm for semiinfinite programming (SIP). For this algorithm, we establish a perturbation theorem under mild conditions. As a corollary of the perturbation theorem, we obtain the global convergence result, that is, any accumulation point of the sequence generated by the algorithm is the solution of SIP. We get this global convergence result without any boundedness condition or coercive condition. Another corollary of the perturbation theorem shows that the perturbation function at zero point is lower semi‐continuous if and only if the algorithm forces the sequence of objective function convergence to the optimal value of SIP. Finally, numerical results are given.

A feasible point adaptation of the Blankenship and Falk algorithm for semi-infinite programming

Discretization methods for semi-infinite programming do not provide a feasible point in a finite number of iterations. We propose a method that computes a feasible point with an objective value better than or equal to a target value f0 or proves that such a point does not exist. Then a binary search on the space of objective values can be performed to obtain a feasible, \({\epsilon}\)-optimal solution. The algorithm is based on the algorithm proposed in (Blankenship JW, Falk JE in J Optim Theory Appl 19(2):261–281, 1976). Under mild assumptions it terminates in a finite number of iterations.

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.

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

This paper presents a smoothing projected Newton-type method for solving the semi-infinite programming (SIP) problem. We first reformulate the KKT system of the SIP problem into a system of constrained nonsmooth equations. Then we solve this system by a smoothing projected Newton-type algorithm. At each iteration only a system of linear equations needs to be solved. The feasibility is ensured via the aggregated constraint under some conditions. Global and local superlinear convergence of this method is established under some standard assumptions. Preliminary numerical results are reported.

A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints

This paper proposes a new algorithm for solving a type of complicated optimal power flow (OPF) problems in power systems, i.e., OPF problems with transient stability constraints (OTS). The OTS is converted into a semi-infinite programming (SIP) via some suitable function analysis. Then based on the KKT system of the reformulated SIP, a smoothing quasi-Newton algorithm is presented in which the numerical integration is used. The convergence of the algorithm is established. An OTS problem in power system is tested, which shows that the proposed algorithm is promising.

A quasi-Newton interior point method for semi-infinite programming

We propose an interior point method for solving nonlinear semi-infinite programming (SIP) problems. The method is based on a transcription of the SIP problem into a nonlinear finite problem. We apply the interior point strategy to the finite problem and obtain an algorithm for nonlinear SIP. The evaluation of the constraint functions and derivatives of the finite problem requires the numerical evaluation of integrals that we compute by an adaptative trapezoid formula. Numerical results with a collection of several problems from the SIPAMPL [A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes. SIPAMPL v2.0: Semi-infinite programming with AMPL. Technical Report ALG/EF/4-2002, Universidade do Minho, Braga, Portugal, December 2002. [http://www.norg.uminho.pt/aivaz/].] database are also shown. †E-mail: emgpf@dps.uminho.pt ‡E-mail: p.gomes@ic.ac.uk

A parallel algorithm for semi-infinite programming

The implementation results of an algorithm designed to solve semi-infinite programming problems are reported. Due to its computational intensity, parallelisation is used in two stages of the algorithm—for evaluating the global optimum and for checking the feasibility of constraints. The algorithms are parallelised using MPI—the message passing interface. The algorithms are then applied to engineering and macroeconomic modelling problems, including designing robust optimally informative experiments for dynamic model identification and verification, and robust macroeconomic policies for inflation targeting.

Numerical experiments in semi-infinite programming

A quasi-Newton algorithm for semi-infinite programming using an L ∞ exact penalty function is described, and numerical results are presented. Comparisons with three Newton algorithms and one other quasi-Newton algorithm show that the algorithm is very promising in practice.

Design of simple robust decouplers with joint stability and performance specifications

The presence of major interactions between manipulated and controlled variables can make it difficult for a control system to achieve the desired level of performance. When such interactions are present, one common practice is to use decoupling control. Although this strategy is straightforward, it can be very sensitive to the process model used. In this paper, a simple robust decoupler is presented. It is developed using structured singular value analysis and by defining a function that quantifies the allowable difference between an “ideal” decoupler and an approximation based on lead-lag elements. The approximate decoupler is determined (i.e. tuned) using a semi-infinite programming algorithm which has rapid convergence characteristics. A case study of decoupling control for a high purity distillation column illustrates the benefits of the proposed technique relative to a short-cut decoupler tuning method.

A purification algorithm for semi-infinite programming

In this paper we present a purification algorithm for semi-infinite linear programming. Starting with a feasible point, the algorithm either finds an improved extreme point or concludes with the unboundedness of the problem. The method is based on the solution of a sequence of linear programming problems. The study of some recession conditions has allowed us to establish a weak assumption for the finite convergence of this algorithm. Numerical results illustrating the method are given.

A projected lagrangian algorithm for semi-infinite programming

A globally convergent algorithm is presented for the solution of a wide class of semi-infinite programming problems. The method is based on the solution of a sequence of equality constrained quadratic programming problems, and usually has a second order convergence rate. Numerical results illustrating the method are given.

Applications of the Hahn-Banach Theorem in Approximation Theory

Applications of the Hahn-Banach Theorem in Approximation Theory