Generalized semi-infinite programming: numerical aspects

Generalized semi-infinite optimization problems (GSIP) are considered. It is investigated how the numerical methods for standard semi-infinite programming (SIP) can be extended to GSIP. Newton methods can be extended immediately. For discretization methods the situation is more complicated. These difficulties are discussed and convergence results for a discretization- and an exchange method are derived under fairly general assumptions on GSIP. The question is answered under which conditions GSIP represents a convex problem.     

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.     

On generalized semi-infinite optimization and bilevel optimization

The paper studies the connections and differences between bilevel problems (BL) and generalized semi-infinite problems (GSIP). Under natural assumptions (GSIP) can be seen as a special case of a (BL). We consider the so-called reduction approach for (BL) and (GSIP) leading to optimality conditions and Newton-type methods for solving the problems. We show by a structural analysis that for (GSIP)-problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general (BL)-problems not. The genericity behavior of (BL) and (GSIP) is in particular studied for linear problems.     

On Optimality Conditions for Generalized Semi-Infinite Programming Problems

Generalized semi-infinite optimization problems (GSIP) are considered. We generalize the well-known optimality conditions for minimizers of order one in standard semi-infinite programming to the GSIP case. We give necessary and sufficient conditions for local minimizers of order one without the assumption of local reduction. The necessary conditions are derived along the same lines as the first-order necessary conditions for GSIP in a recent paper of Jongen, Rückmann, and Stein (Ref. 1) by assuming the so-called extended Mangasarian–Fromovitz constraint qualification. Using the ideas of a recent paper of Rückmann and Shapiro, we give short proofs of necessary and sufficient optimality conditions for minimizers of order one under the additional assumption of the Mangasarian–Fromovitz constraint qualification at all local minimizers of the so-called lower-level problem.     

Generalized semi-infinite programming: A tutorial

This tutorial presents an introduction to generalized semi-infinite programming (GSIP) which in recent years became a vivid field of active research in mathematical programming. A GSIP problem is characterized by an infinite number of inequality constraints, and the corresponding index set depends additionally on the decision variables. There exist a wide range of applications which give rise to GSIP models; some of them are discussed in the present paper. Furthermore, geometric and topological properties of the feasible set and, in particular, the difference to the standard semi-infinite case are analyzed. By using first-order approximations of the feasible set corresponding constraint qualifications are developed. Then, necessary and sufficient first- and second-order optimality conditions are presented where directional differentiability properties of the optimal value function of the so-called lower level problem are used. Finally, an overview of numerical methods is given.     

On generalized semi-infinite programming

This paper surveys some basic properties of the class of generalized semi-infinite programming problems (GSIP) where the infinite index set of inequality constraints depends on the state variables and all emerging functions are assumed to be continuously differentiable. There exists a wide range of applications which can be modelled as a (GSIP). The paper discusses extensions of the Mangasarian-Fromovitz, Kuhn-Tucker and Abadie constraint qualification to (GSIP) and presents related first order optimality conditions of Fritz-John and Karush-Kuhn-Tucker type. By using directional differentiability properties of the optimal value function of the lower level problem, first and second order necessary and sufficient optimality conditions are discussed. Several examples illustrate the results presented. The work of this author was supported by CONACYT (México) under grant 44003.     

Semi-infinite programming method for optimal power flow with transient stability and variable clearing time of faults

This paper presents a new nonlinear programming problem arising in the control of power systems, called optimal power flow with transient stability constraint and variable clearing time of faults and abbreviated as OTS-VT. The OTS-VT model is converted into a implicit generalized semi-infinite programming (GSIP) problem. According to the special box structure of the reformulated GSIP, a solution method based on bi-level optimization is proposed. The research in this paper has two contributions. Firstly, it generalizes the OTS study to general optimal power flow with transient stability problems. From the viewpoint of practical applications, the proposed research can improve the decision-making ability in power system operations. Secondly, the reformulation of OTS-VT also provides a new background and a type of GSIP in the research of mathematical problems. Numerical results for two chosen power systems show that the methodology presented in this paper is effective and promising.     

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.     

A lifting method for generalized semi-infinite programs based on lower level Wolfe duality

This paper introduces novel numerical solution strategies for generalized semi-infinite optimization problems (GSIP), a class of mathematical optimization problems which occur naturally in the context of design centering problems, robust optimization problems, and many fields of engineering science. GSIPs can be regarded as bilevel optimization problems, where a parametric lower-level maximization problem has to be solved in order to check feasibility of the upper level minimization problem. The current paper discusses several strategies to reformulate this class of problems into equivalent finite minimization problems by exploiting the concept of Wolfe duality for convex lower level problems. Here, the main contribution is the discussion of the non-degeneracy of the corresponding formulations under various assumptions. Finally, these non-degenerate reformulations of the original GSIP allow us to apply standard nonlinear optimization algorithms.     

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.     

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.     

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.     

Application of general semi-infinite programming to lapidary cutting problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interior-point method developed by Stein [O. Stein, Bi-level Strategies in Semi-infinite Programming, Kluwer Academic Publishers, Boston, 2003]. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on real-world data are also presented.     

Semi-Infinite Programming and Applications to Minimax Problems

A minimisation problem with infinitely many constraints – semi-infinite programming problem (SIP) is considered. The problem is solved using a two stage procedure that searches for global maximum violation of the constraints. A version of the algorithm that searches for any violation of constraints is also considered, and the performance of the two algorithm is compared. An application to solving minimax problem (with and without coupled constraints) is given and a comparison with the algorithm for continuous minimax of Rustem and Howe (2001) is included. Finally, we consider an application to chemical engineering problems.     

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.     

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.     

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.     

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.     

Duality for semi-definite and semi-infinite programming

In this article, we study semi-definite and semi-infinite programming problems (SDSIP), which includes semi-infinite linear programs and semi-definite programs as special cases. We establish that a uniform duality between the homogeneous (SDSIP) and its Lagrangian-type dual problem is equivalent to the closedness condition of certain cone. Moreover, this closedness condition was assured by a generalized canonically closedness condition and a Slater condition. Corresponding results for the nonhomogeneous (SDSIP) problem were obtained by transforming it into an equivalent homogeneous (SDSIP) problem.     

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...