Interval Methods for Semi-Infinite Programs

A new approach for the numerical solution of smooth, nonlinear semi-infinite programs whose feasible set contains a nonempty interior is presented. Interval analysis methods are used to construct finite nonlinear, or mixed-integer nonlinear, reformulations of the original semi-infinite program under relatively mild assumptions on the problem structure. In certain cases the finite reformulation is exact and can be solved directly for the global minimum of the semi-infinite program (SIP). In the general case, this reformulation is over-constrained relative to the SIP, such that solving it yields a guaranteed feasible upper bound to the SIP solution. This upper bound can then be refined using a subdivision procedure which is shown to converge to the true SIP solution with finite -optimality. In particular, the method is shown to converge for SIPs which do not satisfy regularity assumptions required by reduction-based methods, and for which certain points in the feasible set are subject to an infinite number of active constraints. Numerical results are presented for a number of problems in the SIP literature. The solutions obtained are compared to those identified by reduction-based methods, the relative performances of the nonlinear and mixed-integer nonlinear formulations are studied, and the use of different inclusion functions in the finite reformulation is investigated.     

Lower level duality and the global solution of generalized semi-infinite programs

The reformulation of generalized semi-infinite programs (GSIP) to simpler problems is considered. These reformulations are achieved under the assumption that a duality property holds for the lower level program (LLP). Lagrangian duality is used in the general case to establish the relationship between the GSIP and a related semi-infinite program (SIP). Practical aspects of this reformulation, including how to bound the duality multipliers, are also considered. This SIP reformulation result is then combined with recent advances for the global, feasible solution of SIP to develop a global, feasible point method for the solution of GSIP. Reformulations to finite nonlinear programs, and the practical aspects of solving these reformulations globally, are also discussed. When the LLP is a linear program or second-order cone program, specific duality results can be used that lead to stronger results. Numerical examples demonstrate that the global solution of GSIP is computationally practical via the solution of these duality-based reformulations.     

A Nonsmooth L-M Method for Solving the Generalized Nonlinear Complementarity Problem over a Polyhedral Cone

In this paper the generalized nonlinear complementarity problem (GNCP) defined on a polyhedral cone is reformulated as a system of nonsmooth equations. Based on this reformulation, the famous Levenberg-Marquardt (L-M) algorithm is employed to obtain its solution. Theoretical results that relate the stationary points of the merit function to the solution of the GNCP are presented. Under mild assumptions, we show that the L-M algorithm is both globally and superlinearly convergent. Moreover, a method to calculate a generalized Jacobian is given and numerical experimental results are presented.     

A SQP METHOD FOR GENERAL NONLINEAR COMPLEMENTARITY PROBLEMS

§ 1　 IntroductionThe nonlinear complementarity problem(NCP) is to find a pointx∈Rn such thatx Tf(x) =0 ,x≥ 0 ,f(x)≥ 0 ,(1 .1 )where f is a continuously differentiable function from Rninto itself.It is well known thatthe NCP is equivalent to a system of smoothly nonlinear equations with nonnegative con-straintsH (z)∶ =y -f(x)x . y =0 ,s.t. x≥ 0 ,y≥ 0 ,(1 .2 )where z=(x,y) and x y=(x1 y1 ,...,xnyn) T.Based on the above reformulation,many in-terior-point methods are established;see,fo…     

非线性互补问题的SQP方法

In this paper, the nonlinear complementarity problem is transformed into the least squares problem with nonnegative constraints ,and a SQP algorithm for this reformulation based on a damped Gauss-Newton type method is presented. It is shown that the algorithm is globally and locally superlinearly (quadratically) convergent without the assumption of monotonicity.     

仿射变换内点Levenberg-Marquardt法解KKT系统

提供了一类新的结合非单调内点回代线搜索技术的仿射变换Levenberg-Marquardt法解Karush-Kuhn-Tucker（KKT）系统. 基于由KKT系统转化得到的等价的部分变量具有非负约束的最小化问题,建立了Levenberg-Marquardt方程. 证明了算法不仅具有整体收敛性,而且在合理的假设条件下,算法具有超线性收敛速率. 数值结果验证了算法的实际有效性.     

Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems

In this paper, we propose a Newton-type method for solving a semismooth reformulation of monotone complementarity problems. In this method, a direction-finding subproblem, which is a system of linear equations, is uniquely solvable at each iteration. Moreover, the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem. We show that the algorithm is globally convergent under some mild assumptions. Next, by slightly modifying the direction-finding problem, we propose another Newton-type method, which may be considered a restricted version of the first algorithm. We show that this algorithm has a superlinear, or possibly quadratic, rate of convergence under suitable assumptions. Finally, some numerical results are presented. Supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.     

半无限规划的一阶最优性条件和牛顿型算法

在Fischer-Burmeister非线性互补函数的基础上,得到了半无限规划问题的一个新的一阶必要条件,并将半无限规划问题转化成一个光滑的无约束优化问题,给出了适合该问题的一个Damp-Newton算法,数值例子表明:算法结构简单,数值计算有效.     

A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems

A new algorithm for the solation of large-scale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact Levenberg-Marquardt-type algorithm for its solution. Under mild assumptions, and requiring only the approximate solution of a linear system at each iteration, the algorithm is shown to be both globally and superlinearly convergent, even on degenerate problems. Numerical results for problems with up to 10 000 variables are presented. Partially supported by Agenzia Spaziale Italiana, Roma, Italy.     

A GLOBAL LINEAR AND LOCAL QUADRATIC SINGLE-STEP NONINTERIOR CONTINUATION METHOD FOR MONOTONE SEMIDEFINITE COMPLEMENTARITY PROBLEMS

A noninterior continuation method is proposed for semidefinite complementarity problem (SDCP). This method improves the noninterior continuation methods recently developed for SDCP by Chen and Tseng. The main properties of our method are: (i) it is well defined for the monotones SDCP; (ii) it has to solve just one linear system of equations at each step; (iii) it is shown to be both globally linearly convergent and locally quadratically convergent under suitable assumptions.     

Local reduction based SQP-type method for semi-infinite programs with an infinite number of second-order cone constraints

The second-order cone program (SOCP) is an optimization problem with second-order cone (SOC) constraints and has achieved notable developments in the last decade. The classical semi-infinite program (SIP) is represented with infinitely many inequality constraints, and has been studied extensively so far. In this paper, we consider the SIP with infinitely many SOC constraints, called the SISOCP for short. Compared with the standard SIP and SOCP, the studies on the SISOCP are scarce, even though it has important applications such as Chebychev approximation for vector-valued functions. For solving the SISOCP, we develop an algorithm that combines a local reduction method with an SQP-type method. In this method, we reduce the SISOCP to an SOCP with finitely many SOC constraints by means of implicit functions and apply an SQP-type method to the latter problem. We study the global and local convergence properties of the proposed algorithm. Finally, we observe the effectiveness of the algorithm through some numerical experiments.     

A constrained optimization reformulation and a feasible descent direction method for L_{1/2} regularization

In this paper, we first propose a constrained optimization reformulation to the \(L_{1/2}\) regularization problem. The constrained problem is to minimize a smooth function subject to some quadratic constraints and nonnegative constraints. A good property of the constrained problem is that at any feasible point, the set of all feasible directions coincides with the set of all linearized feasible directions. Consequently, the KKT point always exists. Moreover, we will show that the KKT points are the same as the stationary points of the \(L_{1/2}\) regularization problem. Based on the constrained optimization reformulation, we propose a feasible descent direction method called feasible steepest descent method for solving the unconstrained \(L_{1/2}\) regularization problem. It is an extension of the steepest descent method for solving smooth unconstrained optimization problem. The feasible steepest descent direction has an explicit expression and the method is easy to implement. Under very mild conditions, we show that the proposed method is globally convergent. We apply the proposed method to solve some practical problems arising from compressed sensing. The results show its efficiency.     

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.     

Converging algorithm for a class of BMI applied on state-dependent stabilization of switched systems

In this paper, a design of a stabilizing state-dependent switching law is presented. First, it is based on a new reformulation of this control problem into a class of BMI (Bilinear Matrix Inequality) problem. The originality of this reformulation is that it will be used, in a second part, by an algorithm dedicated to solve this class of BMI without loss of generality. The construction of this algorithm is then specified, and the proof of its convergence is given for the two-subsystem case. The last part of the work consists in presenting the problem for a polytopic switching system, and how the previous algorithm can be extended. The proposed scheme is illustrated by an academical example.     

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 smoothing Newton-type method for solving the <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> spectral estimation problem with lower and upper bounds

This paper discusses the L ₂ spectral estimation problem with lower and upper bounds. To the best of our knowledge, it is unknown if the existing methods for this problem have superlinear convergence property or not. In this paper we propose a nonsmooth equation reformulation for this problem. Then we present a smoothing Newton-type method for solving the resulting system of nonsmooth equations. Global and local superlinear convergence of the proposed method are proved under some mild conditions. Numerical tests show that this method is promising.     

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. Qi’s work is supported by the Hong Kong Research Grant Council. Ling’s work was supported by the Zhejiang Provincial National Science Foundation of China (Y606168). Tong’s work was done during her visit to The Hong Kong Polytechnic University. Her work is supported by the NSF of China (60474070) and the Technology Grant of Hunan (06FJ3038). Zhou’s work is supported by Australian Research Council.     

On finite termination of an iterative method for linear complementarity problems

Based on a well-known reformulation of the linear complementarity problem (LCP) as a nondifferentiable system of nonlinear equations, a Newton-type method will be described for the solution of LCPs. Under certain assumptions, it will be shown that this method has a finite termination property, i.e., if an iterate is sufficiently close to a solution of LCP, the method finds this solution in one step. This result will be applied to a recently proposed algorithm by Harker and Pang in order to prove that their algorithm also has the finite termination property.     

Successive Optimization Method via Parametric Monotone Composition Formulation*

In this paper a successive optimization method for solving inequality constrained optimization problems is introduced via a parametric monotone composition reformulation. The global optimal value of the original constrained optimization problem is shown to be the least root of the optimal value function of an auxiliary parametric optimization problem, thus can be found via a bisection method. The parametric optimization subproblem is formulated in such a way that it is a one-parameter problem and its value function is a monotone composition function with respect to the original objective function and the constraints. Various forms can be taken in the parametric optimization problem in accordance with a special structure of the original optimization problem, and in some cases, the parametric optimization problems are convex composite ones. Finally, the parametric monotone composite reformulation is applied to study local optimality.     

A homotopy interior point method for semi-infinite programming problems

This paper presents a homotopy interior point method for solving a semi-infinite programming (SIP) problem. For algorithmic purpose, based on bilevel strategy, first we illustrate appropriate necessary conditions for a solution in the framework of standard nonlinear programming (NLP), which can be solved by homotopy method. Under suitable assumptions, we can prove that the method determines a smooth interior path from a given interior point to a point w ^*, at which the necessary conditions are satisfied. Numerical tracing this path gives a globally convergent algorithm for the SIP. Lastly, several preliminary computational results illustrating the method are given.