Duality Approach Research Articles

Inside–outside duality with artificial backgrounds

We use the inside–outside duality approach proposed by Kirsch–Lechleiter to identify transmission eigenvalues (TEs) associated with artificial backgrounds. We prove that for well chosen artificial backgrounds, in particular for the ones with zero index of refraction at the inclusion location, one obtains a necessary and sufficient condition characterizing TEs via the spectrum of the modified far field operator. We also complement the existing literature with a convergence result for the invisible generalized incident field associated with the TEs.This work is based on several of the pioneering works of our dearest colleague and friend Armin Lechleiter and is dedicated to his memory.

Migrant enclaves: disempowering economic ghettos or sanctuaries of opportunities for migrants?

Purpose This paper aims to examine the migrant dilemma about operating extensively in migrant enclaves vs integration in host communities. Design/methodology/approach The paper is a critical literature review contrasting views and perspectives of the role of migrant enclaves in migrant integration and contribution in new societies. Research in the area of ethnic enclaves has been polarised: on the one hand, the optimists argue the critical benefits of migrant and ethnic community networks, thus downplaying potential drawbacks of such networks and the disadvantage externally imposed on migrants; on the other hand, the pessimists overemphasise the disadvantages of ethnic enclaves, portraying them as ghettos of alienation. Findings Based on the social solidarity integration model and immigrant-host and social interaction theory, the paper posits that migrant community networks could intentionally or unintentionally engender cultural alienation, worsening an already precarious educational, cultural and economic exclusion. Thus, migrants could remain in lower societal roles and experience limited upward social mobility if they operate exclusively within migrant and ethnic networks. However, ethnic enclaves, at the same time, offer the initial psychological nurturing on which future successful socialisation work with migrant communities can be built. Research limitations/implications From a research angle, the theorisation of migrant enclave requires a new approach, which identifies dynamism and contextualisation as central to the debate. Practical implications From a policy perspective, the research suggests the rethinking of the role of community support systems (and the wider enclave debate). The organisational implications the research suggests a shift of the organisational paradigm in the way migrant organisations manage themselves and support members in the enclave. Originality/value This paper’s contribution is to take a duality approach to studying the ethnic enclave and posits that this will engender effective social policy that helps reduce economic inequality.

Energy‐efficiency fog computing resource allocation in cyber physical internet of things systems

Cyber physical internet of things systems (CPIoTs), taking advantages of cyber physical systems, have been considered as a promising technology to provide better interaction and interoperability among various machines. However, the development of CPIoTs suffers severely from big data. In this context, fog computing is proposed to handle the big data bottleneck of CPIoTs. In this study, the authors focus on the joint optimisation of the communication resources and computation resources in fog computing-based CPIoTs to maximise the overall system energy efficiency, in which multiple fog nodes and end users are taken into consideration. Moreover, since the channel estimation error will become serious with the expanding scale, the imperfect channel state information is considered in this study. The formulated optimisation problem is a mixed integer non-linear problem which is indeed non-deterministic polynomial hard, hence a probability distribution method is proposed to reformulate the problem into a non-probability form, and the resource allocation algorithm based on Dinkelbach algorithm and Lagrange duality approach is adopted to tackle the problem efficiently. The simulation results confirm the effectiveness of the proposed scheme, especially when the scales are enormous.

Constraint-Coupled Distributed Optimization: A Relaxation and Duality Approach

In this paper, we consider a general challenging distributed optimization setup arising in several important network control applications. Agents of a network want to minimize the sum of local cost functions, each one depending on a local variable, subject to local and coupling constraints, with the latter involving all the decision variables. We propose a novel fully distributed algorithm based on a relaxation of the primal problem and an elegant exploration of duality theory. Despite its complex derivation, based on several duality steps, the distributed algorithm has a very simple and intuitive structure. That is, each node finds a primal-dual optimal solution pair of a local relaxed version of the original problem and then updates suitable auxiliary local variables. We prove that agents asymptotically compute their portion of an optimal (feasible) solution of the original problem. This primal recovery property is obtained without any averaging mechanism typically used in dual decomposition methods. To corroborate the theoretical results, we show how the methodology applies to an instance of a distributed model-predictive control scheme in a microgrid control scenario.

Vanishing of certain cuts or residues of loop integrals with higher powers of the propagators

Starting from two-loops, there are Feynman integrals with higher powers of the propagators. They arise from self-energy insertions on internal lines. Within the loop-tree duality approach or within methods based on numerical unitarity one needs (among other things) the residue when a raised propagator goes on-shell. We show that for renormalised quantities in the on-shell scheme these residues can be made to vanish already at the integrand level.

Heavy pentaquark spectroscopy in the diquark model

$QQ^\prime qq\bar q$ pentaquarks are studied in a potential model, under the hypothesis that they are composite objects of two diquarks and one antiquark. The interaction between two colored objects includes two contributions, one based on the $q\bar q$ potential in QCD, computed in the gauge/string duality approach, and another describing the spin-spin interaction. The model has been extended to investigate pentaquarks with different quark content, as $Qqqq\bar q$ and $Qqqq\bar Q$, the latter including the states observed by LHCb, $P_c(4380)^+$ and $P_c(4450)^+$, later updated, with a new data sample, to $P_c(4312)^+$, $P_c(4440)^+$, and $P_c(4457)^+$.

Invariance of critical points under Kelvin transform and multiple solutions in exterior domains of $$\mathbb {R}^2$$ R 2

Let \(\Omega ={\mathbb {R}}^2{\setminus }\overline{B(0,1)}\) be the exterior of the closed unit ball. We prove the existence of extremal constant-sign solutions as well as sign-changing solutions of the following boundary value problem $$\begin{aligned} -\Delta u=a(x) f(u)\ \text{ in } \Omega ,\quad u=0\ \text{ on } \partial \Omega =\partial B(0,1), \end{aligned}$$ where the nonnegative coefficient a satisfies a certain integrability condition. We are looking for solutions in the space \(D^{1,2}_0(\Omega )\) which is the completion of \(C^\infty _c(\Omega )\) with respect to the \(\Vert \nabla \cdot \Vert _{2,\Omega }\)-norm. Unlike in the situation of \({\mathbb {R}}^N\) with \(N\ge 3\), the behavior of solutions in the borderline case \(N=2\) considered here is qualitatively significantly different, such as for example, constant-sign solutions in the borderline case are not decaying to zero at infinity, and instead are bounded away from zero. Our main tool in studying the above problem will be the Kelvin transform. We will first show that the Kelvin transform provides an isometric isomorphism between \(D^{1,2}_0(\Omega )\) and the Sobolev space \(H^1_0(B(0,1))\), which is order-preserving. This allows us to establish a one-to-one mapping between solutions of the problem above and solutions of an associated problem in the bounded domain B(0, 1) of the form: $$\begin{aligned} -\Delta u=b(x) f(u)\ \text{ in } B(0,1),\quad u=0\ \text{ on } \partial B(0,1), \end{aligned}$$ where b satisfies an integrability condition in terms of the coefficient a. This duality approach given via the Kelvin transform allows us to handle nonlinearities under sub, super or asymptotically linear hypotheses.

A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality

We investigate via a conjugate duality approach general nonlinear minmax location problems formulated by means of an extended perturbed minimal time function, necessary and sufficient optimality conditions being delivered together with characterizations of the optimal solutions in some particular instances. A parallel splitting proximal point method is employed in order to numerically solve such problems and their duals. We present the computational results obtained in matlab on concrete examples, successfully comparing these, where possible, with earlier similar methods from the literature. Moreover, the dual employment of the proximal method turns out to deliver the optimal solution to the considered primal problem faster than the direct usage on the latter. Since our technique successfully solves location optimization problems with large data sets in high dimensions, we envision its future usage on big data problems arising in machine learning.

Path monotonicity, consistency and axiomatizations of some weighted solutions

On the domain of cooperative games with transferable utility, we introduce path monotonicity, a property closely related to fairness (van den Brink, in Int J Game Theory 30:309–319, 2001). The principle of fairness states that if a game changes by adding another game in which two players are symmetric, then their payoffs change by the same amount. Under efficiency, path monotonicity is a relaxation of fairness that guarantees that when the worth of the grand coalition varies, the players’ payoffs change according to some monotone path. In this paper, together with the standard properties of projection consistency (Funaki, in Dual axiomatizations of solutions of cooperative games. Mimeo, New York, 1998) and covariance, we show that path monotonicity characterizes the weighted surplus division solutions. Interestingly, replacing projection consistency by either self consistency (Hart and Mas-Colell, in Econometrica 57:589–614, 1989) or max consistency (Davis and Maschler, in Nav Res Logist Q 12:223–259, 1965) we obtain new axiomatic characterizations of the weighted Shapley values and the prenucleolus, respectively. Finally, by the duality approach we provide a new axiomatization of the weighted egalitarian non-separable contribution solutions using complement consistency (Moulin, in J Econ Theory 36:120–148, 1985).

The Dao of foreign policy: Understanding China’s dual strategy in the South China Sea

ABSTRACTIn the South China Sea, China neither implements power-maximizing policy nor engages a peaceful approach. Instead, China implements both coercion and cooperation in pursuing its strategic interest in the disputed area. How can we explain China’s paradoxical behavior? This article claims that the best way to explain China’s policy in the South China Sea is to understand the character of dualism in China’s strategic culture. Following constructivist theory that stresses culture, this article argues that Chinese duality approach in the South China Sea is rooted in the philosophy of Daoism. The symbol of Yin-Yang depicts Chinese assertive as well as cooperative behavior in dealing with the South China Sea dispute. Although from a normative perspective Daoism recognizes pacifism and non-violent behavior, the most important feature of Daoism is the assumption that reality consists of two opposing elements that are mutually embedded.

Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness

In this paper linear stochastic transport and continuity equations with drift in critical $L^{p}$ spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works based on stochastic flows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, these regularity results are then applied to prove uniqueness of weak solutions to linear stochastic continuity and transport equations and certain well-posedness results for the associated stochastic differential equation (sDE) (roughly speaking, existence and uniqueness of flows and their $C^\alpha$ regularity, strong uniqueness for the sDE when the initial datum has diffuse law). Finally, we show two types of examples: on the one hand, we present well-posed sDEs, when the corresponding ODEs are ill-posed, and on the other hand, we give a counterexample in the supercritical case.

The Persuasion Duality

We present a unified duality approach to Bayesian persuasion. The optimal dual variable, interpreted as a price function, is shown to be a super-gradient of the concave closure of the objective function at the prior belief. Under regularity conditions, our general duality result implies known results for the case when the objective function depends only on the expected state. We apply our approach to characterize the optimal signal in the case when the state is two-dimensional.

How Reliable is Duality Theory in Empirical Work?

AbstractThe Neoclassical theory of production establishes a dual relationship between the profit value function of a competitive firm and its underlying production technology. This relationship, commonly referred to as duality theory, has been widely used in empirical work to estimate production parameters such as elasticities and returns to scale. We generate a pseudo‐dataset by Monte Carlo simulations, which, starting from known production parameters, yield a dataset with the main characteristics of U.S. agriculture in terms of unobserved firm heterogeneity, decisions under uncertainty, unexpected production and price shocks, endogenous prices, output and input aggregation, measurement error in variables, and omitted variables. Production parameters are not precisely recovered when performing econometric estimation based on the duality approach, and the elasticity estimates are inaccurate. Deviations of own‐ and cross‐price elasticities from initial median values, given our parameter calibration, range between 6% and 690%, with an average of 90%. Also, own‐price elasticities are as imprecisely recovered as cross‐price elasticities. Sensitivity analysis shows that results still hold for different sources and levels of noise, and sample size used in estimation.

Finite-horizon optimal consumption and investment problem with a preference change

In this paper we propose a duality approach to solving an optimal consumption-portfolio selection problem in which an economic agent can choose (irreversible) the preference change time in finite horizon. We assume that after preference change, the agent's coefficient of relative risk aversion becomes higher than before. We use the martingale method and study the variational inequality or free boundary problem arising from the dual problem. Based on partial differential equation (PDE) theory, we analytically characterize the variational inequality and establish the duality relationship. Specifically, we prove that the agent will change his/her preference as soon as the agent's wealth reaches the threshold of wealth as a function of time.

Optimal investment with random endowments and transaction costs: duality theory and shadow prices

This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the duality approach. As an important application of the duality theorem, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form similar to Czichowsky and Schachermayer (Ann Appl Probab 26(3):1888–1941, 2016) as well as in the usual sense using acceptable portfolios.

Energy Efficiency Optimization With SWIPT in MIMO Broadcast Channels for Internet of Things

Simultaneous wireless information and power transfer (SWIPT) is anticipated to have great applications in 5G communication systems and the Internet of Things. In this paper, we address the energy efficiency (EE) optimization problem for SWIPT multiple-input multiple-output broadcast channel (BC) with time-switching (TS) receiver design. Our aim is to maximize the EE of the system whilst satisfying certain constraints in terms of maximum transmit power and minimum harvested energy per user. The coupling of the optimization variables, namely transmit covariance matrices and TS ratios, leads to an EE problem which is nonconvex, and hence very difficult to solve directly. Hence, we transform the original maximization problem with multiple constraints into a suboptimal min–max problem with a single constraint and multiple auxiliary variables. We propose a dual inner/outer layer resource allocation framework to tackle the problem. For the inner-layer, we invoke an extended SWIPT-based BC-multiple access channel (MAC) duality approach and provide two iterative resource allocation schemes under fixed auxiliary variables for solving the dual MAC problem. A subgradient searching scheme is then proposed for the outer-layer in order to obtain the optimal auxiliary variables. Numerical results confirm the effectiveness of the proposed algorithms and illustrate that significant performance gain in terms of EE can be achieved by adopting the proposed extended BC-MAC duality-based algorithm.

Dynamic convex duality in constrained utility maximization

ABSTRACTIn this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of forward and backward stochastic differential equations (FBSDEs) plus some additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. We also find that the optimal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problem directly.

Shortfall risk through Fenchel duality

In this paper, we propose a Fenchel duality approach to study the minimization problem of the shortfall risk. We consider a general increasing and strictly convex loss function, which may be more general than the situation of convex risk measures usually assumed in the literature. We first translate the associated stochastic optimization problem to an equivalent static optimization problem, and then obtain the explicit structure of the optimal randomized test for both complete and incomplete markets. For the incomplete market case, to the best of our knowledge, we obtain for the first time the explicit randomized test, while previous literature only established the existence through the supermartingale optional decomposition approach. We also solve the shortfall risk minimization problem for an insider through the enlargement of filtrations approach.

Non-linear filtering and optimal investment under partial information for stochastic volatility models

This paper studies the question of filtering and maximizing terminal wealth from expected utility in a partially information stochastic volatility models. The special features is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by a stochastic differential Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filters processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named Kushner-Stratonovich equations. Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.

Portfolio optimization under Solvency II

In the current low interest-rate and highly-regulated environment investing capital efficiently is one of the most important challenges insurance companies face. Certain quantitative parts of regulatory requirements (e.g. Solvency II capital requirements) result in constraints on the investment strategies. This paper mathematically describes the implications of Solvency II constraints on the investment strategies of insurance companies in an expected utility framework with a focus on the market risk module. For this constrained expected utility problem, we define a two-step approach that leads to closed-form approximations for the optimal investment strategies. This proposal circumvents the technical difficulties encountered when applying the convex duality approach or the theory of viscosity solutions. The investment strategies found using the two-step approach can be understood as the optimal investment strategies for constraint problems according to Solvency II. The impact of such constraints on the asset allocation and the performance of these strategies is assessed in a numerical case study.