Approximate Duality Research Articles

Azimuth measurement based on OAM phase spectrum of optical vortices

The optical vortex is a new type of structured light field with a helical wavefront. Because of carrying orbital angular momentum (OAM), the optical vortex can be used to measure the shape and angular orientation of a static object in optical remote sensing. These measurements take advantage of variations in the intensity or phase of the OAM mode. In this paper, we propose a method of target azimuth measurement based on OAM phase spectrum of optical vortices. The OAM phase spectrum consists of helical phase terms in different OAM modes, where the helical phase is obtained by a four-step phase-shifting method. By using the approximate duality relationship between the OAM phase spectrum and the azimuth, we can accurately measure the azimuth of a single target or multiple targets. The resolution of the azimuth measurement is inversely proportional to the OAM scanning range. Experimentally, the target azimuth measurement by employing OAM phase spectrum analysis is realized and the measurement error is less than 2.5°. On this basis, the inverse relationship between the OAM scanning range and azimuth resolution is verified, where the error between experimental and theoretical azimuth resolution is less than 0.3°.

Duals and approximate duals of von Neumann–Schatten p-frames

Abstract The purpose of this paper is to study duals and approximate duals of von Neumann–Schatten p-frames. We present a necessary and sufficient condition for the existence of a dual for a von Neumann–Schatten p-frame. Also, von Neumann–Schatten q-Riesz bases are characterized and it is shown that they share many useful properties with their corresponding notions in Hilbert spaces. Especially, it is focused on the uniqueness of their duals, and some equivalent conditions under which a von Neumann–Schatten p-frame admits a unique dual are derived. Moreover, approximate duality for von Neumann–Schatten p-Bessel sequences is introduced and some of the important properties of approximate duals are obtained, mainly their stability under different kinds of perturbations is considered.

Approximate optimality conditions and approximate duality theorems for nonlinear semi-infinite programming problems with uncertainty data

Approximate optimality conditions and approximate duality theorems for nonlinear semi-infinite programming problems with uncertainty data

Approximate optimality and approximate duality in nonsmooth composite vector optimization

This paper concentrates on studying a nonsmooth composite vector optimization problem (P for brevity). We employ a fuzzy necessary condition for approximate (weakly) efficient solutions of a nonconvex and nonsmooth cone constrained vector optimization problem established in [Choung, T. D. Approximate solutions in nonsmooth and nonconvex cone constrained vector optimization Ann. Oper. Res. (2020), https://doi.org/10.1007/s10479-020-03740-3.] and the a chain rule for generalized differentiation to provide a necessary condition which exhibited in a Fritz-John type for approximate (weakly) efficient solutions of P. Sufficient optimality conditions for approximate (weakly) efficient solutions to P are also provided by means of proposing the use of (strictly) approximately generalized convex composite vector functions with respect to a cone. Moreover, an approximate dual vector problem to P is given and strong and converse duality assertions for approximate (weakly) efficient solutions are proved.

Baryogenesis, magnetogenesis and the strength of anomalous interactions

The production of the hypermagnetic gyrotropy is investigated under the assumption that the gauge coupling smoothly evolves during a quasi-de Sitter phase and then flattens out in the radiation epoch by always remaining perturbative. In the plane defined by the strength of the anomalous interactions and by the rate of evolution of the gauge coupling the actual weight of the pseudoscalar interactions turns out to be always rather modest if major deviations from the homogeneity are to be avoided during the inflationary phase. Even if the gauge power spectra are related by duality only in the absence of anomalous contributions, an approximate duality symmetry constrains the late-time form of the hypermagnetic power spectra. Since the hypermagnetic gyrotropy associated with the modes reentering prior to the phase transition must be released into fermions later on, the portions of the parameter space where the obtained baryon asymmetry is close to the observed value are the most relevant for the present ends. For the same range of parameters the magnetic power spectra associated with the modes reentering after symmetry breaking may even be of the order of a few hundredths of a nG over typical length scales comparable with the Mpc prior to the collapse of the protogalaxy.

Characterizations of robust ε-quasi optimal solutions for nonsmooth optimization problems with uncertain data

ABSTRACT This paper deals with robust ε-quasi optimal solutions for a class of nonsmooth optimization problems with uncertain data. Under some mild assumptions, we first establish, by using robust optimization (i.e. worst-case) approach, approximate optimality conditions for this uncertain nonsmooth optimization problem. Then, we introduce a Mixed-type robust approximate dual problem of this uncertain optimization problem, and explore their relationships. Moreover, using a scalarization method, we derive optimality conditions for robust weakly approximate efficient solutions for an uncertain nonsmooth multiobjective optimization problem. We also obtain approximate duality theorems for the uncertain nonsmooth multiobjective optimization problem.

Approximate Duals, Q-Approximate Duals and Morphisms of Hilbert $$C^*$$-Modules

In this paper, approximate duals and Q-approximate duals of standard Bessel sequences are studied, where Q is a bounded (not necessarily adjointable) operator on a Hilbert $$C^*$$ -module. Particularly, we consider the stability of Q-approximate duals under morphisms of Hilbert $$C^*$$ -modules and also their stability under small perturbations is discussed. Then, we get some properties of (a, m)-approximate duals in Hilbert $$C^*$$ -modules. These approximate duals are defined using the distance between the identity operator and the operator defined by multiplying the Bessel multiplier with symbol m by an element a in the center of the $$C^*$$ -algebra. Moreover, using multipliers for pair Bessel sequences, the notion of (a, m)-approximate duality is generalized for the cases that the sequences are non-Bessel or m is not necessarily a symbol.

Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty

In this paper, we deal with robust approximate quasi optimal solutions for a class of nonlinear semi-infinite programming with data uncertainty in both the objective and constraints. By using a new robust-type subdifferential constraint qualification and some generalized convexity assumptions, we first establish approximate optimality conditions for robust approximate quasi optimal solutions of . Then, we introduce a Mixed-type robust approximate dual problem for and investigate robust approximate duality relations between them. Furthermore, we establish a nonsmooth robust approximate saddle point theorem for an uncertain approximate Lagrangian function associated with .

An operator theory approach to the approximate duality of Hilbert space frames

The purpose of this paper is to state some properties and applications of approximate duals constructed by bounded operators. It is shown that there are close relationships between this kind of approximate duals and some concepts in frame theory such as the closeness of two frames, nearly Parseval and scalable frames. We get some equivalent conditions for a frame to be close to another frame in terms of approximate duals. Also, using approximate duals, some conditions for a frame to be nearly Parseval or scalable are derived. Moreover, some other properties of approximate duals are obtained; in particular, some characterizations of them are stated and some of their applications for Gabor frames and erasures are given, especially the excesses of them are considered.

Fair Packing and Covering on a Relative Scale

Fair resource allocation is a fundamental optimization problem with applications in operations research, networking, and economic and game theory. Research in these areas has led to the general acceptance of a class of $\alpha$-fair utility functions parameterized by $\alpha \in [0, \infty]$. We consider $\alpha$-fair packing---the problem of maximizing $\alpha$-fair utilities under positive linear constraints---and provide a simple first-order method for solving it with relative-error guarantees. The method has a significantly lower convergence time than the state of the art, and to analyze it, we leverage the approximate duality gap technique, which provides an intuitive interpretation of the convergence argument. Finally, we introduce a natural counterpart of $\alpha$-fairness for minimization problems and motivate its usage in the context of fair task allocation. This generalization yields $\alpha$-fair covering problems, for which we provide the first near-linear-time solvers with relative-error guarantees.

QCD phase diagram with chiral imbalance in NJL model: duality and lattice QCD results

In addition to temperature and baryon chemical potential there exist other parameters that are essential in the real quark matter. One of them is isospin asymmetry, which does exist in nature, for example, in the compact stars or in heavy ion collisions. Chiral imbalance, the difference between left- and right-handed quarks, is another phenomenon that could occur in quark matter. We have shown that lattice QCD results support the existence of the approximate duality (found in effective models) in real QCD. The rise of pseudo-critical temperature with increase of chiral chemical potential μ5 (the much debated effect recently, a lot of studies on this contradict each other) has been established in terms of just duality notion, hence reinforce confidence in this result and put it on the considerably more solid ground.

Characterizations of Approximate Duality and Saddle Point Theorems for Nonsmooth Robust Vector Optimization

This article deals with robust approximate Pareto efficient solutions for a class of nonsmooth vector optimization problems with data uncertainty. By using robust optimization and scalarization methodology, we first obtain some nonsmooth approximate optimality conditions for this uncertain optimization problem. Then, we introduce a Wolfe-type robust approximate dual problem for this uncertain optimization problem and obtain some nonsmooth robust approximate duality properties. Moreover, we investigate nonsmooth approximate saddle point theorems of an uncertain approximate vector Lagrangian function associated with this uncertain optimization problem.

On approximate solutions and saddle point theorems for robust convex optimization

This paper provides some new results on robust approximate optimal solutions for convex optimization problems with data uncertainty. By using robust optimization approach (worst-case approach), we first establish necessary and sufficient optimality conditions for robust approximate optimal solutions of this uncertain convex optimization problem. Then, we introduce a Wolfe-type robust approximate dual problem and investigate robust approximate duality relations between them. Moreover, we obtain some robust approximate saddle point theorems for this uncertain convex optimization problem. We also show that our results encompass as special cases some optimization problems considered in the recent literature.

Approximate Duals of $g$-frames and Fusion Frames in Hilbert $C^ast-$modules

In this paper, we study approximate duals of $g$-frames and fusion frames in Hilbert $C^ast-$modules. We get some relations between approximate duals of $g$-frames and biorthogonal Bessel sequences, and using these relations, some results for approximate duals of modular Riesz bases and fusion frames are obtained. Moreover, we generalize the concept of $Q-$approximate duality of $g$-frames and fusion frames to Hilbert $C^ast-$modules, where $Q$ is an adjointable operator, and obtain some properties of this kind of approximate duals.

The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods

We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the algorithm converges. We show that in continuous time enforcement of an invariant that this approximate duality gap decreases at a certain rate exactly recovers a wide range of first-order continuous-time methods. We characterize the discretization errors incurred by different discretization methods, and show how iteration-complexity-optimal methods for various classes of problems cancel out the discretization error. The techniques are illustrated on various classes of problems -- including convex minimization for Lipschitz-continuous objectives, smooth convex minimization, composite minimization, smooth and strongly convex minimization, solving variational inequalities with monotone operators, and convex-concave saddle-point optimization -- and naturally extend to other settings.

On quasi approximate solutions for nonsmooth robust semi-infinite optimization problems

This paper devotes to the quasi ε-solution for robust semi-infinite optimization problems (RSIP) involving a locally Lipschitz objective function and infinitely many locally Lipschitz constraint functions with data uncertainty. Under the fulfillment of robust type Guignard constraint qualification and robust type Kuhn-Tucker constraint qualification, a necessary condition for a quasi ε-solution to problem (RSIP). After introducing the generalized convexity, we give a sufficient optimality for such a quasi ε-solution to problem (RSIP). Finally, we also establish approximate duality theorems in term of Wolfe type which is formulated in approximate form.

Duals and multipliers of controlled frames in Hilbert spaces

In this paper, we introduce and characterize controlled dual frames in Hilbert spaces. We also investigate the relation between bounds of controlled frames and their related frames. Then, we define the concept of approximate duality for controlled frames in Hilbert spaces. Next, we introduce multiplier operators of controlled frames in Hilbert spaces and investigate some of their properties. Finally, we show that the inverse of a controlled multiplier operator is also a controlled multiplier operator under some mild conditions.

An Approach to $\epsilon$-duality Theorems for Nonconvex Semi-infinite Multiobjective Optimization Problems

Using a scheme for solving multiobjective optimization problems via a system of corresponding scalar problems, approximate optimality conditions for a nonconvex semi-infinite multiobjective optimization problem are established. As a new approach, the scheme is developed to study the approximate duality theorems of the problem via a pair of primal-dual scalar problems. Several $\epsilon$-duality theorems are given. Furthermore, the existence theorem for almost quasi weakly $\epsilon$-Pareto solutions of the primal problem, and the existence theorem for quasi weakly $\epsilon$-Pareto solutions of the dual problem are established without any constraint qualification.

On approximate solutions for robust convex semidefinite optimization problems

In this paper, we consider approximate solutions ( $$\epsilon $$ -solutions) for a convex semidefinite programming problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we prove an approximate optimality theorem and approximate duality theorems for $$\epsilon $$ -solutions in robust convex semidefinite programming problem under the robust characteristic cone constraint qualification. Moreover, an example is given to illustrate the obtained results.

Approximate Optimality and Approximate Duality for Quasi Approximate Solutions in Robust Convex Semidefinite Programs

In this paper, we study quasi approximate solutions for a convex semidefinite programming problem in the face of data uncertainty. Using the robust optimization approach (worst-case approach), approximate optimality conditions and approximate duality theorems for quasi approximate solutions in robust convex semidefinite programming problems are explored under the robust characteristic cone constraint qualification. Moreover, some examples are given to illustrate the obtained results.