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.     

Generalized semi-infinite programming: Theory and methods

Generalized semi-infinite optimization problems (GSIP) are considered. The difference between GSIP and standard semi-infinite problems (SIP) is illustrated by examples. By applying the `Reduction Ansatz', optimality conditions for GSIP are derived. Numerical methods for solving GSIP are considered in comparison with methods for SIP. From a theoretical and a practical point of view it is investigated, under which assumptions a GSIP can be transformed into an SIP.     

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.     

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.     

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.     

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.     

Inverse optimization in semi-infinite linear programs

Given the costs and a feasible solution for a finite-dimensional linear program (LP), inverse optimization involves finding new costs that are close to the original ones and render the given solution optimal. Ahuja and Orlin employed the absolute weighted sum metric to quantify distances between costs, and then applied duality to establish that inverse optimization reduces to another finite-dimensional LP. This paper extends this to semi-infinite linear programs (SILPs). A convergent Simplex algorithm to tackle the inverse SILP is proposed.     

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.     

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.     

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.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.     

Copositive Programming via Semi-Infinite Optimization

Copositive programming (CP) can be regarded as a special instance of linear semi-infinite programming (SIP). We study CP from the viewpoint of SIP and discuss optimality and duality results. Different approximation schemes for solving CP are interpreted as discretization schemes in SIP. This leads to sharp explicit error bounds for the values and solutions in dependence on the mesh size. Examples illustrate the structure of the original program and the approximation schemes.     

Optimization reformulations of the generalized Nash equilibrium problem using regularized indicator Nikaidô–Isoda function

In this paper, we extend the literature by adapting the Nikaidô–Isoda function as an indicator function termed as regularized indicator Nikaidô–Isoda function, and this is demonstrated to guarantee existence of a solution. Using this function, we present two constrained optimization reformulations of the generalized Nash equilibrium problem (GNEP for short). The first reformulation characterizes all the solutions of GNEP as global minima of the optimization problem. Later this approach is modified to obtain the second optimization reformulation whose global minima characterize the normalized Nash equilibria. Some numerical results are also included to illustrate the behaviour of the optimization reformulations.     

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.     

Mixed integer reformulations of integer programs and the affine TU-dimension of a matrix

Mathematical Programming - We study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables. Such reformulations can be solved...     

Clark's theorem for semi-infinite convex programs

In 1961, Clark proved that if either the feasible region of a linear program or its dual is nonempty and bounded, then the other is unbounded. Recently, Duffin has extended this result to a convex program and its Lagrangian dual. Moreover, Duffin showed that under this boundedness assumption there is no duality gap. The purpose of this paper is to extend Duffin's results to semi-infinite programs.     

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.     

A pathological semi-infinite program verifying Karlovitz's conjecture

In 1965, Duffin and Karlovitz approximated semi-infinite linear programs by a sequence of linear programs where thenth approximating program minimizes the objective function, subject to the firstn constraints. At that time, Karlovitz conjectured the existence of a semi-infinite convex program without a duality gap, whose approximating programs have a duality gap. The purpose of this note is to provide such an example.The author completed this work while at the University of Illinois at Urbana-Champaign, Illinois.     

Uniform LP duality for semidefinite and semi-infinite programming

Recently, a semidefinite and semi-infinite linear programming problem (SDSIP), its dual (DSDSIP), and uniform LP duality between (SDSIP) and (DSDSIP) were proposed and studied by Li et al. (Optimization 52:507–528, 2003). In this paper, we show that (SDSIP) is an ordinary linear semi-infinite program and, therefore, all the existing results regarding duality and uniform LP duality for linear semi-infinite programs can be applied to (SDSIP). By this approach, the main results of Li et al. (Optimization 52:507–528, 2003) can be obtained easily.     

Primal-dual stability in continuous linear optimization

Any linear (ordinary or semi-infinite) optimization problem, and also its dual problem, can be classified as either inconsistent or bounded or unbounded, giving rise to nine duality states, three of them being precluded by the weak duality theorem. The remaining six duality states are possible in linear semi-infinite programming whereas two of them are precluded in linear programming as a consequence of the existence theorem and the non-homogeneous Farkas Lemma. This paper characterizes the linear programs and the continuous linear semi-infinite programs whose duality state is preserved by sufficiently small perturbations of all the data. Moreover, it shows that almost all linear programs satisfy this stability property.