Gauss quadrature is a well-known method for estimating the integral of a continuous function with respect to a given measure as a weighted sum of the function evaluated at a set of node points. Gauss quadrature is tra...
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Gauss quadrature is a well-known method for estimating the integral of a continuous function with respect to a given measure as a weighted sum of the function evaluated at a set of node points. Gauss quadrature is traditionally developed using orthogonal polynomials. We show that Gauss quadrature can also be obtained as the solution to an infinite-dimensional linear program (LP): minimize the th moment among all nonnegative measures that match the through moments of the given measure. While this infinite-dimensional LP provides no computational advantage in the traditional setting of integration on the real line, it can be used to construct Gauss-like quadratures in more general settings, including arbitrary domains in multiple dimensions.
The feasible set M in general semi-infinite programming (GSIP) need not be closed. This fact is well known. We introduce a natural constraint qualification, called symmetric Mangasarian-Fromovitz constraint qualificat...
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The feasible set M in general semi-infinite programming (GSIP) need not be closed. This fact is well known. We introduce a natural constraint qualification, called symmetric Mangasarian-Fromovitz constraint qualification (Sym-MFCQ). The Sym-MFCQ is a nontrivial extension of the well-known (extended) MFCQ for the special case of semi-infinite programming (SIP) and disjunctive programming. Under the Sym-MFCQ the closure M has an easy and also natural description. As a consequence, we get a description of the interior and boundary of M. The Sym-MFCQ is shown to be generic and stable under C-1-perturbations of the defining functions. For the latter stability the consideration of the closure of M is essential. We introduce an appropriate notion of Karush-Kuhn-Tucker (KKT) points. We show that local minimizers are KKT points under the Sym-MFCQ.
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 con...
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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.
We consider the Scenario Convex Program (SCP) for two classes of optimization problems that are not tractable in general: Robust Convex Programs (RCPs) and Chance-Constrained Programs (CCPs). We establish a probabilis...
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We consider the Scenario Convex Program (SCP) for two classes of optimization problems that are not tractable in general: Robust Convex Programs (RCPs) and Chance-Constrained Programs (CCPs). We establish a probabilistic bridge from the optimal value of SCP to the optimal values of RCP and CCP in which the uncertainty takes values in a general, possibly infinite dimensional, metric space. We then extend our results to a certain class of non-convex problems that includes, for example, binary decision variables. In the process, we also settle a measurability issue for a general class of scenario programs, which to date has been addressed by an assumption. Finally, we demonstrate the applicability of our results on a benchmark problem and a problem in fault detection and isolation.
This paper first introduces an original trajectory model using B-splines and a new semi-infinite programming formulation of the separation constraint involved in air traffic conflict problems. A new continuous optimiz...
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This paper first introduces an original trajectory model using B-splines and a new semi-infinite programming formulation of the separation constraint involved in air traffic conflict problems. A new continuous optimization formulation of the tactical conflict-resolution problem is then proposed. It involves very few optimization variables in that one needs only one optimization variable to determine each aircraft trajectory. Encouraging numerical experiments show that this approach is viable on realistic test problems. Not only does one not need to rely on the traditional, discretized, combinatorial optimization approaches to this problem, but, moreover, local continuous optimization methods, which require relatively fewer iterations and thereby fewer costly function evaluations, are shown to improve the performance of the overall global optimization of this non-convex problem. (C) 2014 Elsevier B.V. All rights reserved.
We introduce a new method for a task of maximal material utilization, which is to fit a flexible, scalable three-dimensional body into another aiming for maximal volume whereas position and shape may vary. The difficu...
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We introduce a new method for a task of maximal material utilization, which is to fit a flexible, scalable three-dimensional body into another aiming for maximal volume whereas position and shape may vary. The difficulty arises from the containment constraint which is not easy to handle numerically. We use a collision detection method to check the constraint and reformulate the problem such that the constraint is hidden within the objective function. We apply methods from parametric optimization to proof that the objective function remains at least continuous. We apply the new approach to the problem of fitting a gemstone into a roughstone. For this previous approaches based on semi-infinite optimization exist, to which we compare our algorithm. The new algorithm is more suitable for necessary global optimization techniques and numerical results show that it works reliably and in general outperforms the previous approaches in both runtime and solution quality.
This paper was originally motivated by the problem of providing a point-based formula (only involving the nominal data, and not data in a neighborhood) for estimating the calmness modulus of the optimal set mapping in...
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This paper was originally motivated by the problem of providing a point-based formula (only involving the nominal data, and not data in a neighborhood) for estimating the calmness modulus of the optimal set mapping in linear semi-infinite optimization under perturbations of all coefficients. With this aim in mind, the paper establishes as a key tool a basic result on finite-valued convex functions in the -dimensional Euclidean space. Specifically, this result provides an upper limit characterization of the boundary of the subdifferential of such a convex function. When applied to the supremum function associated with our constraint system, this characterization allows us to derive an upper estimate for the aimed calmness modulus in linear semi-infinite optimization under the uniqueness of nominal optimal solution.
We discuss numerical reduction methods for an optimal control problem of semi-infinite type with finitely many control parameters but infinitely many constraints. We invoke known a priori error estimates to reduce the...
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We discuss numerical reduction methods for an optimal control problem of semi-infinite type with finitely many control parameters but infinitely many constraints. We invoke known a priori error estimates to reduce the number of constraints. In a first strategy, we apply uniformly refined meshes, whereas in a second more heuristic strategy we use adaptive mesh refinement and provide an a posteriori error estimate for the control based on perturbation arguments.
We address a central theme of empirical model building: the incorporation of first-principles information in a data-driven model-building process. By enabling modelers to leverage all available information, regression...
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We address a central theme of empirical model building: the incorporation of first-principles information in a data-driven model-building process. By enabling modelers to leverage all available information, regression models can be constructed using measured data along with theory-driven knowledge of response variable bounds, thermodynamic limitations, boundary conditions, and other aspects of system knowledge. We expand the inclusion of regression constraints beyond intra-parameter relationships to relationships between combinations of predictors and response variables. Since the functional form of these constraints is more intuitive, they can be used to reveal hidden relationships between regression parameters that are not directly available to the modeler. First, we describe classes of a priori modeling constraints. Next, we propose a semi-infinite programming approach for the incorporation of these novel constraints. Finally, we detail several application areas and provide extensive computational results. (C) 2014 Elsevier Ltd. All rights reserved.
The aim of this article is to give a survey of some basic theory of semi-infinite programming. In particular, we discuss various approaches to derivations of duality, discretization, and first- and second-order optima...
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The aim of this article is to give a survey of some basic theory of semi-infinite programming. In particular, we discuss various approaches to derivations of duality, discretization, and first- and second-order optimality conditions. Some of the surveyed results are well known while others seem to be less noticed in that area of research.
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