Dual nonlinear programming problems in partially ordered Banach spaces

Summary The concept of duality plays an important role in mathematical programming and has been studied extensively in a finite dimensional Eucledian space, (see e.g. [13, 4, 6, 8]). More recently various dual problems with functionals as objective functions have been studied in infinite dimensional vector spaces [5, 7, 1, 10, 12].In this note we consider a nonlinear minimization problem in a partially ordered Banach space. It is assumed that the objective function of this problem is given by a (nonlinear) operator and that its feasible domain is defined by a system of (nonlinear) operator inequalities. In analogy to the finite dimensional case we associate with this minimization problem a dual maximization problem which is defined in the Cartesian product of certain Banach spaces. It is shown that under suitable assumptions the main results of the finite dimensional duality theory can be extended to this general case. This extension is based on optimality conditions obtained in [11].     

Positive extension and completion problems for a class of structured matrices

A class of matrices, defined by a displacement rank property, is introduced. Completion and extension problems are studied for matrices in this class, under certain positivity constraints. The extension problem is reduced to a standard interpolation problem for Schur matrix valued functions.     

Positive extension and completion problems for a class of structured matrices

A class of matrices, defined by a displacement rank property, is introduced. Completion and extension problems are studied for matrices in this class, under certain positivity constraints. The extension problem is reduced to a standard interpolation problem for Schur matrix valued functions.     

A partially observable decision problem under a shifted likelihood ratio ordering

In this paper, we discuss a partially observable sequential decision problem under a shifted likelihood ratio ordering. Since we employ the Bayes' theorem for the learning procedure, we treat this problem under several assumptions. Under these assumptions, we obtain some fundamental results about the relation between prior and posterior information. We also consider an optimal stopping problem for this partially observable Markov decision process.     

Role of fundamental solutions for optimal Lipschitz extensions on hyperbolic space

In this paper we consider the problem of finding the relation between absolutely minimizing Lipschitz extension of a given function defined over a subset of the hyperbolic space and the viscosity solution of the PDE that appears from the associated variational problem. Here we have shown that the absolute minimizers can be fully characterized by a comparison principle (comparison with cones) with the fundamental solutions of the associated PDE. We have finally proved that the three properties, (i) comparison with cones, (ii) absolutely minimizing Lipschitz extension and (iii) viscosity solution of associated PDE, are equivalent.     

On the strong convergence of derivatives in a time optimal problem

We consider a time optimal problem for a system described by a differential inclusion, whose right hand side is not necessarily convex valued. Under the assumption of strict convexity of the map obtained by convexifying the original, non-convex valued map, we obtain the strong convergence of the derivatives of any uniformly converging minimizing sequence. The assumptions required by this result are satisfied, for instance, by the classical brachystochrone problem and by Fermat’s principle.     

Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities

In this paper, we introduce and study a relaxed extragradient method for finding solutions of a general system of variational inequalities with inverse-strongly monotone mappings in a real Hilbert space. First, this system of variational inequalities is proven to be equivalent to a fixed point problem of nonexpansive mapping. Second, by using the demi-closedness principle for nonexpansive mappings, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient method converges strongly to a solution of this system of variational inequalities. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems. J.-C. Yao’s research was partially supported by a grant from the National Science Council.     

The isomorphism problem for uniserial modules over an arbitrary ring

Abstract

First, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined over algebras given by quivers.     

On the proximal point method in Hadamard spaces

Proximal point method is one of the most influential procedure in solving nonlinear variational problems. It has recently been introduced in Hadamard spaces for solving convex optimization, and later for variational inequalities. In this paper, we study the general proximal point method for finding a zero point of a maximal monotone set-valued vector field defined on a Hadamard space and valued in its dual. We also give the relation between the maximality and Minty’s surjectivity condition, which is essential for the proximal point method to be well-defined. By exploring the properties of monotonicity and the surjectivity condition, we were able to show under mild assumptions that the proximal point method converges weakly to a zero point. Additionally, by taking into account the metric subregularity, we obtained the local strong convergence in linear and super-linear rates.     

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.     

Learning AMP chain graphs and some marginal models thereof under faithfulness

This paper deals with chain graphs under the Andersson–Madigan–Perlman (AMP) interpretation. In particular, we present a constraint based algorithm for learning an AMP chain graph a given probability distribution is faithful to. Moreover, we show that the extension of Meek's conjecture to AMP chain graphs does not hold, which compromises the development of efficient and correct score + search learning algorithms under assumptions weaker than faithfulness.We also study the problem of how to represent the result of marginalizing out some nodes in an AMP CG. We introduce a new family of graphical models that solves this problem partially. We name this new family maximal covariance–concentration graphs because it includes both covariance and concentration graphs as subfamilies.     

Monotone Variable–Metric Algorithm for Linearly Constrained Nonlinear Programming

A new method for linearly constrained nonlinear programming is proposed. This method follows affine scaling paths defined by systems of ordinary differential equations and it is fully parallelizable. The convergence of the method is proved for a nondegenerate problem with pseudoconvex objective function. In practice, the algorithm works also under more general assumptions on the objective function. Numerical results obtained with this computational method on several test problems are shown.     

Topological differential fields

We consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields (requiring no interaction between the derivation and the other primitives of the language). We give a criterion under which the expansion still admits a model-completion which we axiomatize. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. As a corollary, we show a transfer result for the NIP property. We also give a geometrical axiomatization of that model-completion. Then, for certain differential valued fields, we extend the positive answer of Hilbert’s seventeenth problem and we prove an Ax-Kochen-Ershov theorem. Similarly, we consider first-order theories of topological fields admitting a model-companion and their expansion to differential fields, and under a similar criterion as before, we show that the expansion still admits a model-companion. This last result can be compared with those of M. Tressl: on one hand we are only dealing with a single derivation whereas he is dealing with several, on the other hand we are not restricting ourselves to definable expansions of the ring language, taking advantage of our topological context. We apply our results to fields endowed with several valuations (respectively several orders).     

A general theorem of existence of quasi absolutely minimal Lipschitz extensions

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier “quasi” to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.     

Some Remarks on the Convex Feasibility Problem and Best Approximation Problem

In this paper we investigate several solution algorithms for the convex fea- sibility problem(CFP)and the best approximation problem(BAP)respectively.The algorithms analyzed are already known before,but by adequately reformulating the CFP or the BAP we naturally deduce the general projection method for the CFP from well-known steepest decent method for unconstrained optimization and we also give a natural strategy of updating weight parameters.In the linear case we show the connec- tion of the two projection algorithms for the CFP and the BAP respectively.In addition, we establish the convergence of a method for the BAP under milder assumptions in the linear case.We also show by examples a Bauschke's conjecture is only partially correct.     

Mean-value at risk portfolio efficiency: approaches based on data envelopment analysis models with negative data and their empirical behaviour

We deal with the problem of an investor who is using a mean-risk model for accessing efficiency of investment opportunities. Our investor employs value at risk on several risk levels at the same time which corresponds to the approach called risk shaping. We review several data envelopment analysis (DEA) models which can deal with negative data. We show that a diversification–consistent extension of the DEA models based on a directional distance measure can be used to identify the Pareto–Koopmans efficient investment opportunities. We derive reformulations as chance constrained, nonlinear and mixed-integer problems under particular assumptions. In the numerical study, we access efficiency of US industry representative portfolios based on empirical distribution of random returns. We employ bootstrap and jackknife to investigate the empirical properties of the efficiency estimators.     

Multi-reservoir production optimization

When a large oil or gas field is produced, several reservoirs often share the same processing facility. This facility is typically capable of processing only a limited amount of commodities per unit of time. In order to satisfy these processing limitations, the production needs to be choked, i.e., scaled down by a suitable choke factor. A production strategy is defined as a vector valued function defined for all points of time representing the choke factors applied to reservoirs at any given time. In the present paper we consider the problem of optimizing such production strategies with respect to various types of objective functions. A general framework for handling this problem is developed. A crucial assumption in our approach is that the potential production rate from a reservoir can be expressed as a function of the remaining recoverable volume. The solution to the optimization problem depends on certain key properties, e.g., convexity or concavity, of the objective function and of the potential production rate functions. Using these properties several important special cases can be solved. An admissible production strategy is a strategy where the total processing capacity is fully utilized throughout a plateau phase. This phase lasts until the total potential production rate falls below the processing capacity, and after this all the reservoirs are produced without any choking. Under mild restrictions on the objective function the performance of an admissible strategy is uniquely characterized by the state of the reservoirs at the end of the plateau phase. Thus, finding an optimal admissible production strategy, is essentially equivalent to finding the optimal state at the end of the plateau phase. Given the optimal state a backtracking algorithm can then used to derive an optimal production strategy. We will illustrate this on a specific example.     

Generalized solutions of quasi variational inequalities

This paper deals with multivalued quasi variational inequalities with pseudo-monotone and monotone maps. The primary objective of this work is to show that the notion of generalized solutions can be employed to investigate multivalued pseudo-monotone quasi variational inequalities. It is a well-known fact that a quasi variational inequality can conveniently be posed as a fixed point problem through the so-called variational selection. For pseudo-monotone maps, the associated variational selection is a nonconvex map, and the fixed point theorems can only be applied under restrictive assumptions on the data of quasi variational inequalities. On the other hand, the generalized solutions are defined by posing a minimization problem which can be solved by a variant of classical Weierstrass theorem. It turns out that far less restrictive assumptions on the data are needed in this case. To emphasis on the strong difference between a classical solution and a generalized solution, we also give a new existence theorem for quasi variational inequalities with monotone maps. The main existence result is proved under a milder coercivity condition. We also relax a few other conditions from the monotone map. Due to its flexibility, it seems that the notion of generalized solutions can be employed to study quasi variational inequalities for other classes of maps as well.     

A relaxation method for solving systems with infinitely many linear inequalities

The problem of finding a feasible solution to a linear inequality system arises in numerous contexts. We consider solving linear semi-infinite inequality systems via an extension of the relaxation method for finite linear inequality systems. The difficulties are discussed and a convergence result is derived under fairly general assumptions on a large class of linear semi-infinite inequality systems.     

On the extension of measures with values in partially ordered semigroups

Clearly the sum as well as the maximum of two real numbers can be presented as a semigroup operation. So the measure with values in a partially ordered semigroup is a common generalization of additive or subadditive and maxitive measures (see Section 4). The extension of such measures we realize by the transfinite induction (see also [2]) and we use a result of [1] for real valued functions.