Duality-based Approach Research Articles

Spherical Artin groups in 2D quantum Hall system framework: a duality-based approach to description particle statistics in a sphere geometry

Spherical Artin groups in 2D quantum Hall system framework: a duality-based approach to description particle statistics in a sphere geometry

Error Estimates for Sparse Optimal Control Problems by Piecewise Linear Finite Element Approximation

Optimization problems with $L^1$-control cost functional subject to an elliptic partial differential equation (PDE) are considered. However, different from the finite dimensional $l^1$-regularization optimization, the resulting discretized $L^1$-norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the $L^1$-norm. It is inevitable that this technique will incur an additional error. Different from the traditional approach, a duality-based approach and an accelerated block coordinate descent (ABCD) method is introduced to solve this type of problem via its dual. Based on the discretized dual problem, a new discretized scheme for the $L^1$-norm is presented. Compared new discretized scheme for $L^1$-norm with the nodal quadrature formula, the advantages of our new discretized scheme can be demonstrated in terms of the approximation order. More importantly, finite element error estimates results for the primal problem with the new discretized scheme for the $L^1$-norm are provided, which confirm that this approximation scheme will not change the order of error estimates.

Hierarchical duality-based planning of transmission networks coordinating active distribution network operation

The active distribution network becomes more stochastic due to high penetration of renewable energy sources, and it can transfer surplus power for transmission needs. If the distribution optimization problem collaborates with the conventional transmission expansion formulation, the coordination can also postpone the investment decisions for the transmission network, not only optimize the operation of the entire power system. In view of these, this paper presents a bi-level hierarchy transmission expansion formulation coordinating distribution networks. The upper level of the hierarchical decision-making framework corresponds to the objective and constraints of the transmission network, whereas the lower level relates to the distribution perspective. An approximation technique is used to make the constraints of the distribution network linear, which guarantees that the proposed bi-level program can be solved using the duality-based approach. Numerical tests carried out on two transmission-distribution integration networks demonstrate the effectiveness and high performance of the proposed hierarchical duality-based model to accommodate renewable energy. In addition, sensitivities of the transmission investment decisions are analysed in terms of externalities in transmission and distribution networks. Compared with conventional isolated optimization, results show that the total costs of T24D9 and T38D3 by the proposed hierarchical coordinated optimization can respectively reduce by 34.38 M$ and 382.23 M$. So the economic benefits are remarkable, and the proposed optimization is reasonable and valuable.

A Duality-Based Approach for Distributed Min–Max Optimization

In this paper, we consider a distributed optimization scenario, motivated by peak-demand minimization, in which a set of processors aims at cooperatively solving a class of min–max optimization problems. The min–max structure and a twofold coupling make the problem challenging in a distributed setup. We propose a distributed algorithm based on the derivation of a series of dual problems and the application of properties from min–max optimization. The resulting distributed algorithm, despite its complex derivation, has a simple structure consisting of a primal optimization and a suitable dual update. We prove the convergence of the proposed algorithm in objective value, and, moreover, that every limit point of the primal sequence is an optimal (thus feasible) solution. This primal recovery property is of key importance in applications since it allows each agent to compute its portion of the global optimal strategy without resorting to any recovery mechanism. Finally, we provide numerical computations for peak-demand optimization in a network of thermostatically controlled loads and show that our algorithm outperforms a plain distributed subgradient performed on the dual.

Energy Storage Siting and Sizing in Coordinated Distribution and Transmission Systems

If energy storage units are installed and operated in a coordinated manner, they can improve efficiency of the transmission and distribution systems. This paper presents a bilevel program to optimally site and size distributed energy storage units in the distribution system and to use them for both distribution and transmission system needs. The upper level of the bilevel program represents the operating objective and constraints of the distribution system, whereas the lower level accounts for the transmission perspective. The bilevel model is solved using the duality-based approach and its performance is tested on the IEEE transmission and distribution benchmarks.

An efficient duality-based approach for PDE-constrained sparse optimization

In this paper, elliptic optimal control problems involving the $L^1$-control cost ($L^1$-EOCP) is considered. To numerically discretize $L^1$-EOCP, the standard piecewise linear finite element is employed. However, different from the finite dimensional $l^1$-regularization optimization, the resulting discrete $L^1$-norm does not have a decoupled form. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the $L^1$-norm. It is clear that this technique will incur an additional error. To avoid the additional error, solving $L^1$-EOCP via its dual, which can be reformulated as a multi-block unconstrained convex composite minimization problem, is considered. Motivated by the success of the accelerated block coordinate descent (ABCD) method for solving large scale convex minimization problems in finite dimensional space, we consider extending this method to $L^1$-EOCP. Hence, an efficient inexact ABCD method is introduced for solving $L^1$-EOCP. The design of this method combines an inexact 2-block majorized ABCD and the recent advances in the inexact symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block. The proposed algorithm (called sGS-imABCD) is illustrated at two numerical examples. Numerical results not only confirm the finite element error estimates, but also show that our proposed algorithm is more efficient than (a) the ihADMM (inexact heterogeneous alternating direction method of multipliers), (b) the APG (accelerated proximal gradient) method.

Optimal Contracts for Intermediaries in Online Advertising

In online advertising, the prevalent method advertisers employ to acquire impressions is to contract with an intermediary. These contracts involve upfront payments made by the advertisers to the intermediary, in exchange for running campaigns on their behalf. This paper studies the optimal contract offered by the intermediary in a setting where advertisers’ budgets and targeting criteria are private. This problem can naturally be formulated as a multidimensional mechanism design problem, which in general is hard to solve. We tackle this problem by combining a performance space characterization technique, which relies on delineating the expected cost and value achievable by any feasible (dynamic) bidding policy, and a duality-based approach, which reduces the optimal contract design problem to a tractable convex optimization problem. This approach yields a crisp characterization of the intermediary’s optimal bidding policy: the policy is stationary and bids a weighted average of the values associated with different types (to guarantee that the advertiser reports her type truthfully) that is appropriately shaded (to account for budget constraints). Additionally, when advertisers have identical value distributions, our formulation yields a closed-form characterization of the optimal contract. Our results indicate that an intermediary can profitably provide bidding service to a budget-constrained advertiser and at the same time increase the overall market efficiency. The online appendix is available at https://doi.org/10.1287/opre.2017.1618 .

A Duality-Based Approach for Distributed Optimization with Coupling Constraints

In this paper we consider a distributed optimization scenario in which a set of agents has to solve a convex optimization problem with separable cost function, local constraint sets and a coupling inequality constraint. We propose a novel 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 solves a local version of the original problem relaxation, and updates suitable dual variables. We prove the algorithm correctness and show its effectiveness via numerical computations.

Active Fault Isolation: A Duality-Based Approach via Convex Programming

This paper presents the mathematical conditions and the associated design methodology of an active fault diagnosis technique for continuous-time linear systems. Given a set of faults known a priori, the system is modeled by a finite family of linear time-invariant systems, accounting for one healthy and several faulty configurations. By assuming bounded disturbances and using a residual generator, an invariant set and its projection in the residual space (i.e., its limit set) are computed for each system configuration. Each limit set, related to a single system configuration, is parameterized with respect to the system input. Thanks to this design, active fault isolation can be guaranteed by the computation of a test input, either constant or periodic, such that the limit sets associated with different system configurations are separated, and the residual converges toward one limit set only. In order to alleviate the complexity of the explicit computation of the limit set, an implicit dual representation is adopted, leading to efficient procedures, based on quadratic programming , for computing the test input. The developed methodology offers a competent continuous-time solution to the optimization-based computation of the test input via Hahn–Banach duality. Simulation examples illustrate the application of the proposed active fault diagnosis methods and its efficiency in providing a solution, even in relatively large state-dimensional problems.

Dynamic costs and moral hazard: A duality-based approach

The marginal cost of effort often increases as effort is exerted. In a dynamic moral hazard setting, dynamically increasing costs create information asymmetry. This paper characterizes the optimal contract and helps explain the popular yet thus far puzzling use of non-linear incentives, for example, in sales-force compensation. The result is obtained by complementing the standard dynamic program with a novel dynamic dual formulation. The dual program is monotonic and sub-modular, providing stronger results, including a proof for the sufficiency of one-shot deviations.

Optimized Resource Management in Heterogeneous Wireless Networks

The dense and pervasive deployment of wireless small cells can boost the performance of existing macrocellular networks; however, it poses significant challenges pertaining to the cross-tier interference management. In this letter, the downlink resource allocation problem for an underlay small cell network is studied. In this network, the protection of the macrocell tier is achieved by imposing cross-tier interference constraints in the resource allocation problem. To solve the underlying mixed-integer resource allocation problem, we propose two different algorithms. The first algorithm is developed by applying the duality-based optimization approach for the relaxed problem, which enables distributed implementation. The second distributed algorithm, which enables coordination is devised based on matching theory. Simulation results show that the proposed duality-based algorithm outperforms the greedy approach by 4% in terms of sum-rate whereas the matching-based algorithm with tier-coordination yields performance gains up to 17% compared with the duality-based approach.

Optimal Contracts for Intermediaries in Online Advertising

In online advertising, the prevalent method advertisers employ to acquire impressions is to contract with an intermediary. These contracts involve upfront payments made by the advertisers to the intermediary, in exchange for running campaigns on their behalf. This paper studies the optimal contract offered by the intermediary in a setting where advertisers' budgets and targeting criteria are private. This problem can naturally be formulated as a multi-dimensional mechanism design problem, which in general is hard to solve. We tackle this problem by combining a performance space characterization technique, which relies on delineating the expected cost and value achievable by any feasible (dynamic) bidding policy, and a duality-based approach, which reduces the optimal contract design problem to a tractable convex optimization problem. This approach yields a crisp characterization of the intermediary's optimal bidding policy: the policy is stationary and bids a weighted average of the values associated with different types (to guarantee that the advertiser reports her type truthfully) that is appropriately shaded (to account for budget constraints). Additionally when advertisers have identical value distributions, our formulation yields a closed-form characterization of the optimal contract. Our results indicate that an intermediary can profitably provide bidding service to a budget-constrained advertiser and at the same time increase the overall market efficiency.

Short-Term Operation Scheduling in Renewable-Powered Microgrids: A Duality-Based Approach

This paper considers the operation scheduling problem in renewable-powered microgrids, which is used to determine the least-cost unit commitment (UC) and the associated dispatch, while meeting load, environmental, and system operating requirements. The intermittency nature of the renewable energy sources, as well as microgrid's capacity to operate either in parallel with, or autonomously of, the traditional power grid, pose new challenges to this classic optimization task. A probability-based concept, probability of self-sufficiency (PSS), is introduced to indicate the probability that the microgrid is capable of meeting local demand in a self-sufficient manner. Furthermore, to the best of our knowledge, we make the first attempt in approaching the mixed-integer UC problem from a convex optimization perspective, which leads to an analytical closed-form characterization of the optimal commitment and dispatch solutions. The simulation results show that 1) the proposed method achieves an efficient performance that incurs no loss of optimality with lower complexity than existing algorithms; 2) an energy storage system (ESS) with suitable capacity contributes to the self-sufficiency target of a microgrid, and the stored energy varies less remarkably as the microgrid tends to operate more independently; 3) the proposed method provides guidelines in deciding the ESS size to achieve a desired PSS.

무선망에서의 상하향 링크 쌍대성 성질을 활용한 분산적 수율 최대화 기법

We consider the throughput-maximization problem for both the up- and downlink in a wireless network with interference channels. For this purpose, we design an iterative and distributive uplink algorithm based on Lagrangian relaxation. Using the uplink power prices and network duality, we achieve throughput-maximization in the dual downlink that has a symmetric channel and an equal power budget compared to the uplink. The network duality we prove here is a generalized version of previous research [10], [11]. Computational tests show that the performance of the up- and downlink throughput for our algorithms is close to the optimal value for the channel orthogonality factor, (0.5, 1]. On the other hand, when the channels are slightly orthogonal ((0, 0.5]), we observe some throughput degradation in the downlink. We have extended our analysis to the real downlink that has a nonsymmetric channel and an unequal power budget compared to the uplink. It is shown that the modified duality-based approach is thoroughly applied to the real downlink. Considering the complexity of the algorithms in [6] and [18], we conclude that these results are quite encouraging in terms of both performance and practical applicability of the generalized duality theorem.

Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach

This paper is devoted to the optimization problem of continuous multi-partitioning, or multi-labeling, which is based on a convex relaxation of the continuous Potts model. In contrast to previous efforts, which are tackling the optimal labeling problem in a direct manner, we first propose a novel dual model and then build up a corresponding duality-based approach. By analyzing the dual formulation, sufficient conditions are derived which show that the relaxation is often exact, i.e. there exists optimal solutions that are also globally optimal to the original nonconvex Potts model. In order to deal with the nonsmooth dual problem, we develop a smoothing method based on the log-sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal-dual model and suggests labelings with maximum entropy. Such a smoothing method for the dual model also yields a new thresholding scheme to obtain approximate solutions. An expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization. Numerical experiments also show that our method outperforms several competitive approaches in various aspects, such as lower energies and better visual quality.

A duality-based approach to elliptic control problems in non-reflexive Banach spaces

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1 (Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with measures and functions of bounded variation as controls are considered. Existence and uniqueness of the corresponding predual problems are discussed, as is the solution of the optimality systems by a semismooth Newton method. Numerical examples illustrate the structural differences in the optimal controls in these Banach spaces, compared to those obtained in corresponding Hilbert space settings.

On linear programming and matrix scaling over the algebraic numbers

Adler and Beling considered the linear programming problem over the real algebraic numbers. They obtained the necessary bounds in terms of a notion of size and dimension needed to justify its polynomial-time solvability, using the ellipsoid method and under some models of computation. Based on a better notion of size than that used by Adler and Beling, we first reduce the feasibility problem in linear programming to some canonical problems preserving its size and its constraint-matrix dimensions. For these canonical problems as well as for the matrix scaling problem, shown to be a more general problem than linear programming, we obtain the necessary bounds; demonstrate simple rounding schemes; justify the applicability of two polynomial-time interior-point algorithms under some models of computation; describe a method for solving a system of linear equations over the algebraic numbers which is a subroutine within these interior-point algorithms under an input model; and give an alternative method to the traditional duality-based approach for the conversion of a general linear programming problem into a feasibility problem.