Lagrangian Dual Problem Research Articles

A Convex Primal Formulation for Convex Hull Pricing

In certain electricity markets, because of non-convexities that arise from their operating characteristics, generators that follow the independent system operator's (ISO's) decisions may fail to recover their cost through sales of energy at locational marginal prices. Discriminatory side payments are made by the ISO to incentivize the compliance of generators. Convex hull pricing is a uniform pricing scheme that minimizes these side payments. The Lagrangian dual problem of the unit commitment problem has been used to determine convex hull prices. However, this approach is computationally expensive. In this paper, we propose a polynomially-solvable primal formulation for the Lagrangian dual problem. This formulation explicitly describes for each generating unit the convex hull of its feasible set and the convex envelope of its cost function. We cast our formulation as a second-order cone program when the cost functions are quadratic, and a linear program when the cost functions are piecewise linear. A 96-period 76-unit transmission-constrained example is solved in less than fifteen seconds on a personal computer.

A Novel Manifold Regularized Online Semi-supervised Learning Model

In the process of human learning, training samples are often obtained successively. Therefore, many human learning tasks exhibit online and semi-supervision characteristics, that is, the observations arrive in sequence and the corresponding labels are presented very sporadically. In this paper, we propose a novel manifold regularized model in a reproducing kernel Hilbert space (RKHS) to solve the online semi-supervised learning (OS2L) problems. The proposed algorithm, named Model-Based Online Manifold Regularization (MOMR), is derived by solving a constrained optimization problem. Different from the stochastic gradient algorithm used for solving the online version of the primal problem of Laplacian support vector machine (LapSVM), the proposed algorithm can obtain an exact solution iteratively by solving its Lagrange dual problem. Meanwhile, to improve the computational efficiency, a fast algorithm is presented by introducing an approximate technique to compute the derivative of the manifold term in the proposed model. Furthermore, several buffering strategies are introduced to improve the scalability of the proposed algorithms and theoretical results show the reliability of the proposed algorithms. Finally, the proposed algorithms are experimentally shown to have a comparable performance to the standard batch manifold regularization algorithm.

Scalable algorithm for the dynamic reconfiguration of the distribution network using the Lagrange relaxation approach

This paper proposes a new methodology for the dynamic reconfiguration of the distribution network (DRDN) which is based on the Lagrange relaxation approach. The aim of DRDN is to determine the optimal topologies (configurations) of the distribution network over the specified time interval. The objective is to minimize the active power losses, subject to the following constraints: branch power flow capacities, allowed ranges of bus voltages, radial network configuration, and limited number of switching (open/close) operations for all switching devices. The paper first introduces the “path-switch-to-switch” approach for the modelling of distribution networks, which is used to formulate DRDN as the mixed integer linear programming (MILP) problem. Then, the specified MILP problem is solved using the Lagrange relaxation approach in two-step procedure. In the first step, the associated Lagrange dual problem is solved, which is created by relaxing the switching operation constraints. The Lagrange dual problem is decoupled and much easier to solve than the original problem. In the second step, the solution of the Lagrange dual problem is used to perform the heuristic search, providing the suboptimal, though feasible solution of the original problem. Finally, the presented DRDN model is extended to multi-objective formulation, which also includes the impact of the network reliability and the switching costs to the DRDN process. The robustness and scalability of the developed algorithm (for application in large-scale distribution networks) are demonstrated with two test examples: 15-bus test benchmark and 1021-bus real-world test system.

A subgradient approach for constrained binary optimization via quantum adiabatic evolution

Outer approximation method has been proposed for solving the Lagrangian dual of a constrained binary quadratic programming problem via quantum adiabatic evolution in the literature. This should be an efficient prescription for solving the Lagrangian dual problem in the presence of an ideally noise-free quantum adiabatic system. However, current implementations of quantum annealing systems demand methods that are efficient at handling possible sources of noise. In this paper, we consider a subgradient method for finding an optimal primal–dual pair for the Lagrangian dual of a constrained binary polynomial programming problem. We then study the quadratic stable set (QSS) problem as a case study. We see that this method applied to the QSS problem can be viewed as an instance-dependent penalty-term approach that avoids large penalty coefficients. Finally, we report our experimental results of using the D-Wave 2X quantum annealer and conclude that our approach helps this quantum processor to succeed more often in solving these problems compared to the usual penalty-term approaches.

Computation of Time Optimal Control Problems Governed by Linear Ordinary Differential Equations

In this paper, a novel numerical algorithm is presented to compute the optimal time of a time optimal control problem where the governing system is a linear ordinary differential equation. By the equivalence between time optimal control problem and norm optimal control problem, computation of the optimal time can be obtained by solving a sequence of norm optimal control problems, which are transferred into their Lagrangian dual problems. The nonsmooth structure of the dual problem is approximated by the iteratively reweighted least square strategy. Several numerical tests are given to show the efficiency of the proposed algorithm.

A comparison of alternative c-conjugate dual problems in infinite convex optimization

In this work, we obtain a Fenchel–Lagrange dual problem for an infinite dimensional optimization primal one, via perturbational approach and using a conjugation scheme called c-conjugation instead of classical Fenchel conjugation. This scheme is based on the generalized convex conjugation theory. We analyse some inequalities between the optimal values of Fenchel, Lagrange and Fenchel–Lagrange dual problems and we establish sufficient conditions under which they are equal. Examples where such inequalities are strictly fulfilled are provided. Finally, we study the relations between the optimal solutions and the solvability of the three mentioned dual problems.

A novel Lagrangian relaxation level approach for scheduling steelmaking-refining-continuous casting production

A Lagrangian relaxation (LR) approach was presented which is with machine capacity relaxation and operation precedence relaxation for solving a flexible job shop (FJS) scheduling problem from the steelmaking-refining-continuous casting process. Unlike the full optimization of LR problems in traditional LR approaches, the machine capacity relaxation is optimized asymptotically, while the precedence relaxation is optimized approximately due to the NP-hard nature of its LR problem. Because the standard subgradient algorithm (SSA) cannot solve the Lagrangian dual (LD) problem within the partial optimization of LR problem, an effective deflected-conditional approximate subgradient level algorithm (DCASLA) was developed, named as Lagrangian relaxation level approach. The efficiency of the DCASLA is enhanced by a deflected-conditional epsilon-subgradient to weaken the possible zigzagging phenomena. Computational results and comparisons show that the proposed methods improve significantly the efficiency of the LR approach and the DCASLA adopting capacity relaxation strategy performs best among eight methods in terms of solution quality and running time.

Rescheduling optimization of steelmaking-continuous casting process based on the Lagrangian heuristic algorithm

This study investigates a challenging problem of rescheduling a hybrid flow shop in the steelmaking-continuous casting (SCC) process, which is a major bottleneck in the production of iron and steel. In consideration of uncertain disturbance during SCC process, we develop a time-indexed formulation to model the SCC rescheduling problem. The performances of the rescheduling problem consider not only the efficiency measure, which includes the total weighted completion time and the total waiting time, but also the stability measure, which refers to the difference in the number of operations processed on different machines for the different stage in the original schedule and revised schedule. With these objectives, this study develops a Lagrangian heuristic algorithm to solve the SCC rescheduling problem. The algorithm could provide a realizable termination criterion without having information about the problem, such as the distance between the initial iterative point and the optimal point. This study relaxes machine capacity constraints to decompose the relaxed problem into charge-level subproblems that can be solved using a polynomial dynamic programming algorithm. A heuristic based on the solution of the relaxed problem is presented for obtaining a feasible reschedule. An improved efficient subgradient algorithm is introduced for solving Lagrangian dual problems. Numerical results for different events and problem scales show that the proposed approach can generate high-quality reschedules within acceptable computational times.

Aspects of optimization with stochastic dominance

We consider stochastic optimization problems with integral stochastic order constraints. This problem class is characterized by an infinite number of constraints indexed by a function space of increasing concave utility functions. We are interested in effective numerical methods and a Lagrangian duality theory. First, we show how sample average approximation and linear programming can be combined to provide a computational scheme for this problem class. Then, we compute the Lagrangian dual problem to gain more insight into this problem class.

Stochastic recursive inclusion in two timescales with an application to the Lagrangian dual problem

In this paper we present a framework to analyze the asymptotic behavior of two timescale stochastic approximation algorithms including those with set-valued mean fields. This paper builds on the works of Borkar and Perkins & Leslie. The framework presented herein is more general as compared to the synchronous two timescale framework of Perkins & Leslie, however the assumptions involved are easily verifiable. As an application, we use this framework to analyze the two timescale stochastic approximation algorithm corresponding to the Lagrangian dual problem in optimization theory.

Ergodic, primal convergence in dual subgradient schemes for convex programming, II: the case of inconsistent primal problems

Consider the utilization of a Lagrangian dual method which is convergent for consistent convex optimization problems. When it is used to solve an infeasible optimization problem, its inconsistency will then manifest itself through the divergence of the sequence of dual iterates. Will then the sequence of primal subproblem solutions still yield relevant information regarding the primal program? We answer this question in the affirmative for a convex program and an associated subgradient algorithm for its Lagrange dual. We show that the primal–dual pair of programs corresponding to an associated homogeneous dual function is in turn associated with a saddle-point problem, in which—in the inconsistent case—the primal part amounts to finding a solution in the primal space such that the Euclidean norm of the infeasibility in the relaxed constraints is minimized; the dual part amounts to identifying a feasible steepest ascent direction for the Lagrangian dual function. We present convergence results for a conditional varepsilon -subgradient optimization algorithm applied to the Lagrangian dual problem, and the construction of an ergodic sequence of primal subproblem solutions; this composite algorithm yields convergence of the primal–dual sequence to the set of saddle-points of the associated homogeneous Lagrangian function; for linear programs, convergence to the subset in which the primal objective is at minimum is also achieved.

Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation

One of the challenging problems in collaborative position localization arises when the distance measurements contain non-line-of-sight (NLOS) biases. Convex optimization has played a major role in modelling such problems and numerical algorithm developments. One of the successful examples is the semi-definite programming (SDP), which translates Euclidean distances into the constraints of positive semidefinite matrices, leading to a large number of constraints in the case of NLOS biases. In this paper, we propose a new convex optimization model that is built upon the concept of Euclidean distance matrix (EDM). The resulting EDM optimization has an advantage that its Lagrangian dual problem is well structured and hence is conducive to algorithm developments. We apply a recently proposed 3-block alternating direction method of multipliers to the dual problem and tested the algorithm on some real as well as simulated data of large scale. In particular, the EDM model significantly outperforms the existing SDP model and several others.

Minimizing value-at-risk in single-machine scheduling

The vast majority of the machine scheduling literature focuses on deterministic problems in which all data is known with certainty a priori. In practice, this assumption implies that the random parameters in the problem are represented by their point estimates in the scheduling model. The resulting schedules may perform well if the variability in the problem parameters is low. However, as variability increases accounting for this randomness explicitly in the model becomes crucial in order to counteract the ill effects of the variability on the system performance. In this paper, we consider single-machine scheduling problems in the presence of uncertain parameters. We impose a probabilistic constraint on the random performance measure of interest, such as the total weighted completion time or the total weighted tardiness, and introduce a generic risk-averse stochastic programming model. In particular, the objective of the proposed model is to find a non-preemptive static job processing sequence that minimizes the value-at-risk (VaR) of the random performance measure at a specified confidence level. We propose a Lagrangian relaxation-based scenario decomposition method to obtain lower bounds on the optimal VaR and provide a stabilized cut generation algorithm to solve the Lagrangian dual problem. Furthermore, we identify promising schedules for the original problem by a simple primal heuristic. An extensive computational study on two selected performance measures is presented to demonstrate the value of the proposed model and the effectiveness of our solution method.

Optimal Precoding for a QoS Optimization Problem in Two-User MISO-NOMA Downlink

In this letter, based on the non-orthogonal multiple access (NOMA) concept, a quality-of-service optimization problem for two-user multiple-input-single-output broadcast systems is considered, given a pair of target interference levels. The minimal power and the optimal precoding vectors are obtained by considering its Lagrange dual problem and via Newton’s iterative algorithm, respectively. Moreover, the closed-form expressions of the minimal transmission power for some special cases are also derived. One of these cases is termed quasi-degraded, which is the key point and will be discussed in detail in this letter. Our analysis further figures out that the proposed NOMA scheme can approach nearly the same performance as optimal dirty paper coding, as verified by computer simulations.

Planar Pose Graph Optimization: Duality, Optimal Solutions, and Verification

Pose graph optimization (PGO) is the problem of estimating a set of poses from pairwise relative measurements. PGO is a nonconvex problem and, currently, no known technique can guarantee the computation of a global optimal solution. In this paper, we show that Lagrangian duality allows computing a globally optimal solution under conditions that are satisfied in most robotics applications and enables to certify optimality of a given estimate. Our first contribution is to frame planar PGO in the complex domain. This makes analysis easier and allows drawing connections with existing literature on unit gain graphs. The second contribution is to formulate and analyze the properties of the Lagrangian dual problem in the complex domain. Our analysis shows that the duality gap is connected to the number of zero eigenvalues of the penalized pose graph matrix. We prove that if this matrix has a single zero eigenvalue, then 1) the duality gap is zero, 2) the primal PGO problem has a unique solution (up to an arbitrary roto-translation), and 3) the primal solution can be computed by scaling an eigenvector of the penalized pose graph matrix. The third contribution is algorithmic: We leverage duality to devise and algorithm that computes the optimal solution when the penalized matrix has a single zero eigenvalue. We also propose a suboptimal variant when the zero eigenvalues are multiple. Finally, we show that duality provides computational tools to verify if a given estimate (e.g., computed using iterative solvers) is globally optimal. We conclude the paper with an extensive numerical analysis. Empirical evidence shows that, in the vast majority of cases (100% of the tests under noise regimes of practical robotics applications), the penalized pose graph matrix has a single zero eigenvalue; hence, our approach allows computing (or verifying) the optimal solution.

Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.

Per-Relay Power Minimization for Multi-user Multi-channel Cooperative Relay Beamforming

We investigate the optimal relay beamforming problem for multi-user peer-to-peer communication with amplify-and-forward relaying in a multi-channel system. Assuming each source–destination (S–D) pair is assigned an orthogonal channel, we formulate the problem as a min–max per-relay power minimization problem with minimum signal-to-noise (SNR) guarantees. After showing that strong Lagrange duality holds for this nonconvex problem, we transform its Lagrange dual problem to a semi-definite programming problem and obtain the optimal relay beamforming vectors. We identify that the optimal solution can be obtained in three cases, depending on the values of the optimal dual variables. These cases correspond to whether the minimum SNR requirement at each S–D pair is met with equality, and whether the power consumption at a relay is the maximum among relays at optimality. We obtain a semi-closed form solution structure of relay beam vectors, and propose an iterative approach to determine relay beam vector for each S–D pair. We further show that the reverse problem of maximizing the minimum SNR with per-relay power budgets can be solved using our proposed algorithm with an iterative bisection search. Through simulation, we analyze the effect of various system parameters on the performance of the optimal solution. Furthermore, we investigated the effect of imperfect channel side information of the second hop on the performance and quantify the performance loss due to either channel estimation error or limited feedback.

On the Optimization of Distributed Compression in Multirelay Cooperative Networks

In this paper, we consider multirelay cooperative networks for the Rayleigh fading channel, where each relay, upon receiving its own channel observation, independently compresses it and forwards the compressed information to the destination. Although the compression at each relay is distributed using Wyner–Ziv coding, there exists an opportunity for jointly optimizing compression at multiple relays to maximize the achievable rate. Considering Gaussian signaling, a primal optimization problem is formulated accordingly. We prove that the primal problem can be solved by resorting to its Lagrangian dual problem, and an iterative optimization algorithm is proposed. The analysis is further extended to a hybrid scheme, where the employed forwarding scheme depends on the decoding status of each relay. The relays that are capable of successful decoding perform a decode-and-forward (DF) scheme, and the rest conduct distributed compression. The hybrid scheme allows the cooperative network to adapt to the changes of the channel conditions and benefit from an enhanced level of flexibility. Numerical results from both spectrum and energy efficiency perspectives show that the joint optimization improves efficiency of compression and identify the scenarios where the proposed schemes outperform the conventional forwarding schemes. The findings provide important insights into the optimal deployment of relays in a realistic cellular network.

Lagrangean relaxation approach to joint optimization for production planning and scheduling of synchronous assembly lines

This paper focuses on simultaneous optimisation of production planning and scheduling problem over a time period for synchronous assembly lines. Differing from traditional top-down approaches, a mixed integer programming model which jointly considers production planning and detailed scheduling constraints is formulated, and a Lagrangian relaxation method is developed for the proposed model, whereby the integrated problem is decomposed into planning, batch sequencing, tardiness and earliness sub-problems. The scheduling sub-problem is modelled as a time-dependent travelling salesman problem, which is solved using a dynasearch algorithm. A proposition of Lagrangian multipliers is established to accelerate the convergence speed of the proposed algorithm. The average direction strategy is employed to solve the Lagrangian dual problem. Test results demonstrate that the proposed model and algorithm are effective and efficient.

The Study of Cross-layer Optimization for Wireless Rechargeable Sensor Networks Implemented in Coal Mines.

Wireless sensor networks deployed in coal mines could help companies provide workers working in coal mines with more qualified working conditions. With the underground information collected by sensor nodes at hand, the underground working conditions could be evaluated more precisely. However, sensor nodes may tend to malfunction due to their limited energy supply. In this paper, we study the cross-layer optimization problem for wireless rechargeable sensor networks implemented in coal mines, of which the energy could be replenished through the newly-brewed wireless energy transfer technique. The main results of this article are two-fold: firstly, we obtain the optimal relay nodes’ placement according to the minimum overall energy consumption criterion through the Lagrange dual problem and KKT conditions; secondly, the optimal strategies for recharging locomotives and wireless sensor networks are acquired by solving a cross-layer optimization problem. The cyclic nature of these strategies is also manifested through simulations in this paper.