Approximate Duality Research Articles

Broadband suppression of backscattering at optical frequencies using low permittivity dielectric spheres

The exact suppression of backscattering from rotationally symmetric objects requires dual symmetric materials where εr = μr. This prevents their design at many frequency bands, including the optical one, because magnetic materials are not available. Electromagnetically small non-magnetic spheres of large permittivity offer an alternative. They can be tailored to exhibit balanced electric and magnetic dipole polarizabilities a1 = b1, which result in approximate zero backscattering. In this case, the effect is inherently narrowband. Here, we put forward a different alternative that allows broadband functionality: Wavelength-sized spheres made from low permittivity materials. The effect occurs in a parameter regime where approximate duality is met for all multipolar order an ≈ bn, in a weakly wavelength dependence fashion. In addition, and despite of the low permittivity, the overall scattering response of these spheres is still significant. Scattering patterns are shown to be highly directive across an octave spanning band. The effect is analytically and numerically shown using the Mie coefficients.

Approximate Lagrangian duality and saddle point optimality in set optimization

In this paper, we establish approximate Lagrangian multiplier rule, Lagrangian duality and saddle point optimality for set optimization problem where the solutions are defined using set relations introduced by Kuroiwa (Kuroiwa D., The natural criteria in set-valued optimization. Surikaisekikenkyusho Kokyuroku 1031 (1998) 85–90).

Weak sharp solutions of mixed variational inequalities in Banach spaces

In this paper, using the approximate duality mapping, we introduce the definition of weak sharpness of the solution set to a mixed variational inequality in Banach spaces. In terms of the primal gap function associated to the mixed variational inequality, we give several characterizations of the weak sharpness.

On quasi $$\epsilon $$ ϵ -solution for robust convex optimization problems

This paper devotes to the quasi \(\epsilon \)-solution (one sort of approximate solutions) for a robust convex optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish approximate optimality theorem and approximate duality theorems in term of Wolfe type on quasi \(\epsilon \)-solution for the robust convex optimization problem. Moreover, some examples are given to illustrate the obtained results.

Duality related to approximate proper solutions of vector optimization problems

In this work we introduce two approximate duality approaches for vector optimization problems. The first one by means of approximate solutions of a scalar Lagrangian, and the second one by considering $$(C,\varepsilon )$$(C,?)-proper efficient solutions of a recently introduced set-valued vector Lagrangian. In both approaches we obtain weak and strong duality results for $$(C,\varepsilon )$$(C,?)-proper efficient solutions of the primal problem, under generalized convexity assumptions. Due to the suitable limit behaviour of the $$(C,\varepsilon )$$(C,?)-proper efficient solutions when the error $$\varepsilon $$? tends to zero, the obtained duality results extend and improve several others in the literature.

From Affine to Two-Source Extractors via Approximate Duality

We establish a new connection between affine and two-source extractors by presenting black-box constructions of two-source extractors for min-entropy rate below half from any affine extractor for min-entropy rate below half. Two such constructions are presented, and one of our constructions can reach arbitrarily small min-entropy rate assuming that the affine extractor has sufficiently good parameters. The first part of our analysis shows that our constructions are two-source dispersers which are weak (but nontrivial) kinds of two-source extractors, also known as “bipartite Ramsey graphs.” To strengthen this result and obtain two-source extractors we introduce the approximate duality conjecture (ADC) and initiate its study. The ADC leads to a rather general result that can be used to convert a natural class of two-source dispersers---``low-rank dispersers''---into two-source extractors. More specifically, we first prove a special case of ADC that implies that the constructions mentioned above are two-source extractors with large (but nontrivial) constant error. In an attempt to reduce the error in our constructions we show that the polynomial Freiman--Ruzsa conjecture (PFR) in additive combinatorics implies a stronger “approximate duality” statement (and that this stronger statement also implies a weak but as-of-yet-unknown version of PFR). This stronger statement implies in turn that our constructions are two-source extractors with exponentially small error.

Approximate duals and nearly Parseval frames

In this paper we introduce approximate duality of g-frames in Hilbert $C^\ast$-modules and we show that approximate duals of g-frames in Hilbert $C^\ast$-modules share many useful properties with those in Hilbert spaces. Moreover, we obtain some new results for approximate duality of frames and g-frames in Hilbert spaces; in particular, we consider approximate duals of $\varepsilon$-nearly Parseval and $\varepsilon$-close frames.

An Additive Combinatorics Approach Relating Rank to Communication Complexity

Identifying complexity measures that bound the communication complexity of a {0,1}-valued matrix M is one the most fundamental problems in communication complexity. Mehlhorn and Schmidt [1982] were the first to suggest matrix-rank as one such measure. Among other things, they showed log rank F(M) CC(M) rankF2(M), where CC ( M ) denotes the (deterministic) communication complexity of the function associated with M , and the rank on the left-hand side is over any field F and on the right-hand side it is over the two-element field F 2. For certain matrices M , communication complexity equals the right-hand side, and this completely settles the question of “communication complexity vs. F 2-rank”. Here we reopen this question by pointing out that, when M has an additional natural combinatorial property---high discrepancy with respect to distributions which are uniform over submatrices---then communication complexity can be sublinear in F 2-rank. Assuming the Polynomial Freiman-Ruzsa (PFR) conjecture in additive combinatorics, we show that CC(M) O(rank F2(M)/log rank F2(M)) for any matrix M which satisfies this combinatorial property. We also observe that if M has low rank over the reals, then it has low rank over F 2 and it additionally satisfies this combinatorial property. As a corollary, our results also give the first (conditional) sublinear bound on communication complexity in terms of rank over the reals, a result improved later by Lovett [2014]. Our proof is based on the study of the “approximate duality conjecture” which was suggested by Ben-Sasson and Zewi [2011] and studied there in connection to the PFR conjecture. First, we improve the bounds on approximate duality assuming the PFR conjecture. Then, we use the approximate duality conjecture (with improved bounds) to get our upper bound on the communication complexity of low-rank matrices.

Approximate duality of g-frames in Hilbert spaces

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.

Optimal Selection of Imperfect Tests for Fault Detection and Isolation

In this paper, we propose new formulations for the problem of test selection in the presence of imperfect tests in order to minimize the total costs of tests subject to lower bound constraints on fault detection and fault isolation. Our formulation allows tests to have multiple outcomes and delays caused by fault propagation, reporting, and transmission. Since the test selection problem is NP-hard even in the presence of perfect binary tests with no delays, we propose genetic algorithm (GA) and Lagrangian relaxation algorithm (LRA) to solve this problem. GA is a general approach for solving the problem with imperfect tests, including the scenarios with delayed and multiple test outcomes. The LRA is suitable for problems with perfect tests, including multiple outcomes. A key advantage of the LRA approach is that it provides an approximate duality gap, which is an upper bound measure of suboptimality of the solution. Our formulations and algorithms are tested on various real-world and simulated systems, and comparisons are made with previous test selection methods developed for perfect tests with no delays. The results show that our methods can efficiently solve the imperfect test selection problem. In addition, they have better performance (measured in terms of the number of tests used) than the methods in the literature for the perfect test selection cases. Finally, the GA has better computational efficiency than the LRA for all of the scenarios with perfect tests.

Dynamic Set-Covering for Real-Time Multiple Fault Diagnosis With Delayed Test Outcomes

The set-covering problem is widely used to model many real-world applications. In this paper, we formulate a generalization of set-covering, termed dynamic set-covering (DSC), which involves a series of coupled set-covering problems over time. We motivate the DSC problem from the viewpoint of a dynamic multiple fault diagnosis problem, wherein faults, possibly intermittent, evolve over time; the fault-test dependencies are deterministic (components associated with passed tests cannot be suspected to be faulty and at least one of the components associated with failed tests is faulty), and the test outcomes may be observed with delay. The objective of the DSC problem is to infer the most probable time sequence of a parsimonious set of failure sources that explains the observed test outcomes over time. The DSC problem is NP-hard and intractable due to the fault-test dependency matrix that couples the failed tests and faults via the constraint matrix, and the temporal dependence of failure sources over time. By relaxing the coupling constraints using Lagrange multipliers, the DSC problem can be decoupled into independent subproblems, one for each fault. Each subproblem is solved using the Viterbi decoding algorithm, and a primal feasible solution is constructed by modifying the Viterbi solutions via a heuristic. The Lagrange multipliers are updated using a subgradient method. The proposed Viterbi-Lagrangian relaxation algorithm provides a measure of suboptimality via an approximate duality gap. As a major practical extension of the above problem, we also consider the problem of diagnosing faults with delayed test outcomes, termed delay DSC. A detailed experimental evaluation of the algorithms is provided using real-world problems that exhibit masking faults.

Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

An elementary h-route flow, for an integer h≥1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h≤3: The size of a minimum h-route cut is at least f/h and at most O(log4 k⋅f) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h=3 that has an approximation ratio of O(log4 k). Previously, polylogarithmic approximation was known only for h-route cuts for h≤2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem.

A texture model based on a concentration of measure

Cartoon-texture regularization is a technique for reconstructing fine-scaledetails from ill-posed imaging problems, which are often ill-posed because ofsome non-invertible convolution kernel. Here we proposea cartoon-texture regularization for deblurring problems with semi-known kernel.The cartoon component is modeled by a function of bounded variation, whilethe texture component is measured by an approximate duality with Lipschitz functions ($W^{1,\infty}$). To approximate the dual Lipschitz norm, which is difficult to calculate, wepropose an approach using concentration of measure. This providesan accurate and differentiable expression. We also present numerical resultsfor our cartoon-texture decomposition, both in the case of a semi-known deblurringkernel and the case of a known kernel. The texture norm enables one tonumerically reconstruct fine scale details that are typically difficult to recover fromblurred images, and for semi-known deblurring the method quickly leads tothe correct kernel, even when that kernel contains noise.

ε-Mixed type duality for nonconvex multiobjective programs with an infinite number of constraints

Using a scalarization method, approximate optimality conditions of a multiobjective nonconvex optimization problem which has an infinite number of constraints are established. Approximate duality theorems for mixed duality are given. Results on approximate duality in Wolfe type and Mond-Weir type are also derived. Approximate saddle point theorems of an approximate vector Lagrangian function are investigated.

Approximate Solutions and Duality Theorems for Continuous-Time Linear Fractional Programming Problems

This article proposes a practical computational procedure to solve a class of continuous-time linear fractional programming problems by designing a discretized problem. Using the optimal solutions of proposed discretized problems, we construct a sequence of feasible solutions of continuous-time linear fractional programming problem and show that there exists a subsequence that converges weakly to a desired optimal solution. We also establish an estimate of the error bound. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.

Approximate duality for vector quasi-equilibrium problems and applications

In this paper, we introduce new versions of ϵ -dual problems of a vector quasi-equilibrium problem with set-valued maps, and we give an ϵ -duality result between approximate solutions of the primal and dual problems. As the first application of the main result, we obtain an ϵ -duality for a vector quasi-equilibrium problem whose ϵ -solutions are understood in the sense of proper efficiency. The second application is devoted to an ϵ -duality for a vector optimization problem with set-valued maps.

STRING THEORY AND DUALITIES IN THE QUANTUM DISSIPATIVE HOFSTADTER SYSTEM

We study the dualities of the quantum dissipative Hofstadter system which describes particles moving in two dimensions, subject to a uniform magnetic field, a periodic potential and a dissipative force. Using the string theory formulation, we show that the system has two kinds of dualities. The duality, previously known as the exact duality in the literature is shown to correspond to a subgroup of the T-dual symmetry group unbroken by the periodic boundary potential in string theory. The other duality is a particle–kink duality in the noncommutative open string theory which is a generalized Schmid duality in the presence of the uniform magnetic field. The kinks of the dissipative Hofstadter model are found to be noncommutative objects. The particle–kink duality, which is called previously the approximate duality, is shown to be also exact. In contrast to the previous derivation, which is based on the Coulomb gas expansion of the partition function and asserts that the duality holds only approximately in the regime of strong magnetic field, the string theory formulation proves that duality holds always exactly in the off-critical regions where the periodic potential becomes strong, regardless of the strength of the magnetic field. The dualities of the DHM may also be useful for studying the rolling tachyon in string theory in the presence of the Neveu–Schwarz (NS) B-field, since both DHM and rolling tachyon in the presence of NS B-field are described by the same action.

Dynamic Multiple Fault Diagnosis: Mathematical Formulations and Solution Techniques

Imperfect test outcomes, due to factors such as unreliable sensors, electromagnetic interference, and environmental conditions, manifest themselves as missed detections and false alarms. This paper develops near-optimal algorithms for dynamic multiple fault diagnosis (DMFD) problems in the presence of imperfect test outcomes. The DMFD problem is to determine the most likely evolution of component states, the one that best explains the observed test outcomes. Here, we discuss four formulations of the DMFD problem. These include the deterministic situation corresponding to perfectly observed coupled Markov decision processes to several partially observed factorial hidden Markov models ranging from the case where the imperfect test outcomes are functions of tests only to the case where the test outcomes are functions of faults and tests, as well as the case where the false alarms are associated with the nominal (fault free) case only. All these formulations are intractable NP-hard combinatorial optimization problems. Our solution scheme can be viewed as a two-level coordinated solution framework for the DMFD problem. At the top (coordination) level, we update the Lagrange multipliers (coordination variables, dual variables) using the subgradient method. At the bottom level, we use a dynamic programming technique (specifically, the Viterbi decoding or Max-sum algorithm) to solve each of the subproblems, one for each component state sequence. The key advantage of our approach is that it provides an approximate duality gap, which is a measure of the suboptimality of the DMFD solution. Computational results on real-world problems are presented. A detailed performance analysis of the proposed algorithm is also discussed.

Approximate Duality

We extend the Lagrangian duality theory for convex optimization problems to incorporate approximate solutions. In particular, we generalize well-known relationships between minimizers of a convex optimization problem, maximizers of its Lagrangian dual, saddle points of the Lagrangian, Kuhn---Tucker vectors, and Kuhn---Tucker conditions to incorporate approximate versions. As an application, we show how the theory can be used for convex quadratic programming and then apply the results to support vector machines from learning theory.

Erratum: Static strings in Randall-Sundrum scenarios and the quark-antiquark potential [Phys. Rev. D73, 106006 (2006)

We calculate the energy of a static string in an AdS slice between two D3-branes with orbifold condition. The energy for configurations with endpoints on a brane grows linearly for large separation between these points. The derivative of the energy has a discontinuity at some critical separation. Choosing a particular position for one of the branes we find configurations with smooth energy. In the limit where the other brane goes to infinity the energy has a Coulombian behaviour for short separations and can be identified with the Cornell potential for a quark anti-quark pair. This identification leads to effective values for the AdS radius, the string tension and the position of the infrared brane. These results suggest an approximate duality between static strings in an AdS slice and a heavy quark anti-quark configuration in a confining gauge theory.