Constrained Extremum Research Articles

A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems

In the paper, we consider the exact minimax penalty function method used for solving a general nondifferentiable extremum problem with both inequality and equality constraints. We analyze the relationship between an optimal solution in the given constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function under the assumption of convexity of the functions constituting the considered optimization problem (with the exception of those equality constraint functions for which the associated Lagrange multipliers are negative—these functions should be assumed to be concave). The lower bound of the penalty parameter is given such that, for every value of the penalty parameter above the threshold, the equivalence holds between the set of optimal solutions in the given extremum problem and the set of minimizers in its associated penalized optimization problem with the exact minimax penalty function.

SADDLE POINT CRITERIA AND THE EXACT MINIMAX PENALTY FUNCTION METHOD IN NONCONVEX PROGRAMMING

A new characterization of the exact minimax penalty function method is presented. The exactness of the penalization for the exact minimax penalty function method is analyzed in the context of saddle point criteria of the Lagrange function in the nonconvex differentiable optimization problem with both inequality and equality constraints. Thus, new conditions for the exactness of the exact minimax penalty function method are established under assumption that the functions constituting considered constrained optimization problem are invex with respect to the same function $\eta $ (exception with those equality constraints for which the associated Lagrange multipliers are negative - these functions should be assumed to be incave with respect to the same function $\eta $). The threshold of the penalty parameter is given such that, for all penalty parameters exceeding this treshold, the equivalence holds between a saddle point of the Lagrange function in the considered constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function.

A Unified Approach for Constrained Extremum Problems: Image Space Analysis

In this paper, by exploiting the image space analysis we investigate a class of constrained extremum problems, the constraining function of which is set-valued. We show that a (regular) linear separation in the image space is equivalent to the existence of saddle points of Lagrangian and generalized Lagrangian functions and we also give Lagrangian type optimality conditions for the class of constrained extremum problems under suitable generalized convexity and compactness assumptions. Moreover, we consider an exact penalty problem for the class of constrained extremum problems and prove that it is equivalent to the existence of a regular linear separation under suitable generalized convexity and compactness assumptions.

Nonlinear Separation Approach to Constrained Extremum Problems

In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.

A multi-scale framework for CFD modelling of multi-phase complex systems based on the EMMS approach

The averaged conservative equations in CFD modeling are inadequate to achieve a complete description of the multi-scale structures in multiphase complex systems. By considering the relationship between meso-scale structures and meso-scale energy consumption, stability conditions mathematically expressed as a mutually constrained extremum are proposed in the Energy-Minimization Multi-Scale (EMMS) approach and indispensable to reflect the compromise of different dominant mechanisms for various multiphase systems. The approach is first applied to global systems to predict and physically interpret the macro-scale structure evolution, i.e., regime transition. Then when applied to computational cells, it corrects interphase momentum transfer and greatly improves the accuracy of coarse-grid CFD simulation.

A Smoothing Method for Zero–One Constrained Extremum Problems

In this paper, the zero–one constrained extremum problem is reformulated as an equivalent smooth mathematical program with complementarity constraints (MPCC), and then as a smooth ordinary nonlinear programming problem with the help of the Fischer–Burmeister function. The augmented Lagrangian method is adopted to solve the resulting problem, during which the non-smoothness may be introduced as a consequence of the possible inequality constraints. This paper incorporates the aggregate constraint method to construct a uniform smooth approximation to the original constraint set, with approximation controlled by only one parameter. Convergence results are established, showing that under reasonable conditions the limit point of the sequence of stationary points generated by the algorithm is a strongly stationary point of the original problem and satisfies the second order necessary conditions of the original problem. Unlike other penalty type methods for MPCC, the proposed algorithm can guarantee that the limit point of the sequence is feasible to the original problem.

On Generalized Constrained Optimization and Separation Theorems

In this paper a generalized format for a constrained extremum problem is considered. Subsequently, the paper investigates and deepens some aspects concerning the linear separation between two sets in the Euclidean space, that are a convex cone and a generic set. A condition equivalent to their linear separation is given. Moreover, a condition equivalent to regular linear separation is proposed; this condition includes also the nonconvex case and it is finalized to the application to the generalized constrained extremum problems.

On Generalized Constrained Optimization and Separation Theorems

In this paper a generalized format for a constrained extremum problem is considered. Subsequently, the paper investigates and deepens some aspects concerning the linear separation between two sets in the Euclidean space, that are a convex cone and a generic set. A condition equivalent to their linear separation is given. Moreover, a condition equivalent to regular linear separation is proposed; this condition includes also the nonconvex case and it is finalized to the application to the generalized constrained extremum problems.

A Geometric Comparison of the Delta and Fieller Confidence Intervals

The comparison of the Delta and Fieller confidence intervals for the ratio of parameters estimated by normally distributed random variables has long been of interest. Our contribution is the construction of a common geometric representation of both the Delta and Fieller intervals defined by two related constrained extrema problems that are subject to a common constraint. The diagrammatic solution to these problems can be used to examine how earlier comparisons based on alternate analytic relationships and simulations have resulted in the particular conclusions they report. We find that the degree to which the Delta and Fieller intervals coincide depends not only on the univariate statistics for the estimated parameters, but also on the agreement of the sign of their estimated correlation and the ratio of estimated parameters. This article has supplementary material online.

On Regularity for Constrained Extremum Problems. Part 2: Necessary Optimality Conditions

A particular theorem for linear separation between two sets is applied in the image space associated with a constrained extremum problem. In this space, the two sets are a convex cone, depending on the constraints (equalities and inequalities) of the given problem and the homogenization of its image. It is proved that the particular linear separation is equivalent to the existence of Lagrangian multipliers with a positive multiplier associated with the objective function (i.e., a necessary optimality condition). A comparison with the constraint qualifications and the regularity conditions existing in the literature is performed.

On Regularity for Constrained Extremum Problems. Part 1: Sufficient Optimality Conditions

The main aspect of the paper consists in the application of a particular theorem of separation between two sets to the image associated with a constrained extremum problem. In the image space, the two sets are a convex cone, which depends on the constraints (equalities or inequalities) of the given problem, and its image. In this way, a condition for the existence of a regular saddle point (i.e., a sufficient optimality condition) is obtained. This regularity condition is compared with those existing in the literature.

Constrained Extremum Problems with Infinite-Dimensional Image: Selection and Necessary Conditions

This paper deals with image space analysis for constrained extremum problems having an infinite-dimensional image. It is shown that the introduction of selection for point-to-set maps and of quasi multipliers allows one to establish optimality conditions for problems where the classical approach fails.

Constrained extremum problems with infinite dimensional image. Selection and saddle point

The paper deals with Image Space Analysis for constrained extremum problems having infinite dimensional image. It is shown that the introduction of selection for point- to-set maps and of quasi-multipliers allows one to establish sufficient optimality conditions for problems, where the classic ones fail.

Chen's inequality in the Lagrangian case

In the theory of submanifolds, the following problem is fundamental: es- tablish simple relationships between the main intrinsic invariants and the main extrinsic invariants of submanifolds. The basic relationships discovered until now are inequalities. To analyze such problems, we follow the idea of C. Udriste that the method of constrained extremum is a natural way to prove geometric inequalities. We improve Chen's inequal- ity which characterizes a totally real submanifold of a complex space form. For that we suppose that the submanifold is Lagrangian and we formulate and analyze a suitable constrained extremum problem.

Extrema of Young’s modulus for cubic and transversely isotropic solids

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young’s modulus E( n ), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E( n ) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young’s modulus on direction n are given as well.

An Optimization-Based Approach to the Multiple Static Delivery Technique in Radiation Therapy

The paper considers the intensity modulated radiation therapy (inverse) treatment planning. An approach to determine the trajectories of the leaves of the multileaf collimator (MLC) in order to produce the prescribed intensity distribution is developed. The paper concentrates on the multiple static delivery technique. A mathematical model for calculating the intensity distribution with the help of locations of the leafheads of subsequent subfields is constructed. Furthermore, an optimization model in which the decision variables are the locations of leafheads is developed. The relevant constraints are considered as well. The optimization problem is a large dimensional constrained nonlinear global extremum problem. It is solved by the LGO (Lipschitz (Continuous) Global Optimizer) program system. Comparisons with other optimization method (Hooke–Jeeves iteration) are included. Numerical experiments are presented to confirm the functionality of the method.

Variational criterion for dissipative structures dominated by coupled mechanisms

By analyzing the similarity between the viscosity-inertia compromise for single fluid in turbulent pipe flow and the particle-fluid compromise in gas-solid fluidization, it is deduced that the variational criterion for dissipative structures dominated jointly by two sub-mechanisms (for instance, viscous effect and inertial effect in turbulent flow; particle movement and fluid movement in fluidization) could be physically expressed as mutually constrained extremum between the individual extremum tendencies of the sub-mechanisms, and could thus be mathematically formulated as a two-objective optimization problem.

Systems With Infinitesimal Mobility: Part II—Compound and Higher-Order Infinitesimal Mechanisms

Building on the analytical results presented in Part I, the issue of multiple singularity is explored. The attained understanding is applied to the analysis and synthesis of compound infinitesimal mechanisms illustrated in several examples. Employing power expansion techniques, higher-order mechanisms are investigated, revealing, in particular, the existence of even-order mechanisms and their peculiar nature. The equivalence condition is established between prestressability and kinematic immobility of underconstrained structural systems with one singularity. A previously unidentified class of systems—unprestressable infinitesimal mechanisms—is revealed. It complements the conventional source of infinitesimally mobile systems associated with a strict constrained extremum and the resulting feature of prestressability.

A necessary optimality condition for nondifferentiable constrained extremum problems

The purpose of this paper is to establish a necessary optimality condition for a certain class of nondifferentiable extremum problems. The analysis is developed via image-space approach and the result generalizes, in finite-dimensional spaces, the classical results concerning locally Lipschitz mathematical programming.

Theorems of the alternative for multifunctions with applications to optimization: General results

Theorems of the alternative and separation theorems have been shown to be very useful concepts in constrained extremum problems (see, for instance, Refs. 1–12). Their use has stressed the concept of image of a constrained extremum problem, which has turned out to be a powerful and promising tool for investigating the main aspects of optimization (see Refs. 13 and 19). It should be pointed out that, in this approach, a finite-dimensional image problem can be associated to the given extremum problem, even if this is infinite-dimensional and provided that its constraints are expressed by functionals. Such a development can be carried on by means of theorems of the alternative for systems of single-valued functions. In this paper, theorems of the alternative for systems of multifunctions are studied, some general properties are stated, and connections with known results investigated. It is shown how the present approach can be used to analyze extremum problems, where the image of the domain of the constraining functions belongs to a functional space. Such a development will be carried on in a subsequent paper.