Lagrangian Duality Theory Research Articles

Optimization of nonhierarchically decomposed problems

The increasing computational power available to practitioners leads to challenging applications of optimization approaches to large scale systems. To address such problems, decomposition of the original or “all-in-one” (AiO) problem into smaller and simpler subproblems is the approach taken by engineers. Analytical target cascading (ATC), a hierarchical decomposition and coordination approach, is extended to model and coordinate problems with nonhierarchical interactions among the subproblems. Convergence results for ATC based on Lagrangian duality theory are extended for the new approach. Under certain conditions, the optimal solution of the AiO problem can be achieved by independently solving the nonhierarchically interacting subproblems. A mathematical example with several subproblems interacting in a network is included and new applications in engineering design are highlighted.

Coordinated beamforming design using duality theory with dynamic cooperation clusters

Uplink–downlink duality has emerged as an attractive approach to optimise the downlink beamforming problem with fixed cooperation clusters where either all base stations serve all terminals or each base station serves only its own terminals. Although easily implementable for co-located base stations, the performance is still limited by out-of-cluster interference. To address these concerns, this study establishes an uplink–downlink duality for the multi-cell multi-user system with dynamic cooperation clusters where each base station has responsibility for the interference leaked to a set of terminals while only serving a subset of them with data. The multi-cell downlink problem of minimising the total transmit power subject to individual signal-to-interference-and-noise ratio requirements under per-base station power constraints is solved via a dual uplink problem. Conditions for beamforming optimality and the optimal downlink beamforming design are derived using Lagrange duality theory. The convergence behaviour of proposed algorithm is shown. The percentage of power saved by proposed algorithm is calculated subject to the user-specific SINR value achieved by zero-forcing (ZF), maximum-ratio transmit (MRT), virtual SINR (VSINR) and layered virtual SINR (LVSINR) under different cooperation scenarios and the sum rate performance is compared.

Multi-period portfolio optimization for asset–liability management with bankrupt control

In this paper, we investigate a multi-period portfolio optimization problem for asset–liability management of an investor who intends to control the probability of bankruptcy before reaching the end of an investment horizon. We formulate the problem as a generalized mean–variance model that incorporates bankrupt control over intermediate periods. Based on the Lagrangian multiplier method, the embedding technique, the dynamic programming approach and the Lagrangian duality theory, we propose a method to solve the model. A numerical example is given to demonstrate our method and show the impact of bankrupt control and market parameters on the optimal portfolio strategy.

Selfish Distributed Compression Over Networks: Correlation Induces Anarchy

We consider the min-cost multicast problem (under network coding) with multiple correlated sources where each terminal wants to losslessly reconstruct all the sources. We study the inefficiency brought forth by the selfish behavior of the terminals in this scenario by modeling it as a noncooperative game among the terminals. The degradation in performance due to the lack of regulation is measured by the Price of Anarchy (POA), which is defined as the ratio between the cost of the worst possible Wardrop equilibrium and the socially optimum cost. Our main result is that in contrast with the case of independent sources, the presence of source correlations can significantly increase the price of anarchy. Toward establishing this result, we first characterize the socially optimal flow and rate allocation in terms of four intuitive conditions. Next, we show that the Wardrop equilibrium is a socially optimal solution for a different set of (related) cost functions. Using this, we construct explicit examples that demonstrate that the POA >; 1 and determine near-tight upper bounds on the POA as well. The main techniques in our analysis are Lagrangian duality theory and the usage of the supermodularity of conditional entropy.

Smooth and Semismooth Newton Methods for Constrained Approximation and Estimation

In the article, we show that the constrained L 2 approximation problem, the positive polynomial interpolation, and the density estimation problems can all be reformulated as a system of smooth or semismooth equations by using Lagrange duality theory. The obtained equations contain integral functions of the same form. The differentiability or (strong) semismoothness of the integral functions and the Hölder continuity of the Jacobian of the integral function were investigated. Then a globalized Newton-type method for solving these problems was introduced. Global convergence and numerical tests for estimating probability density functions with wavelet basis were also given. The research in this article not only strengthened the theoretical results in literatures but also provided a possibility for solving the probability density function estimation problem by Newton-type method.

Duality in vector optimization via augmented Lagrangian

This paper is devoted to developing augmented Lagrangian duality theory in vector optimization. By using the concepts of the supremum and infimum of a set and conjugate duality of a set-valued map on the basic of weak efficiency, we establish the interchange rules for a set-valued map, and propose an augmented Lagrangian function for a vector optimization problem with set-valued data. Under this augmented Lagrangian, weak and strong duality results are given. Then we derive sufficient conditions for penalty representations of the primal problem. The obtained results extend the corresponding theorems existing in scalar optimization.

Power‐efficient resource allocation with QoS guarantees for TDMA fading channels

ABSTRACTThis paper proposes two power‐efficient resource allocation policies with statistical delay Quality of Service (QoS) guarantees for uplink time‐division multiple access (TDMA) communication links. Specifically, the first policy aims at maximizing the system throughput while fulfilling the delay QoS and average power constraints, and the second policy is devised as an effort to minimize the total average power subject to individual delay QoS constraints. Convex optimization problems associated with the resource allocation policies are formulated based on a cross‐layer framework, where the queue at the data link layer is served by the resource allocation policy. By employing the Lagrangian duality theory and the dual decomposition theory, two subgradient iteration algorithms are developed to obtain the globally optimal solutions. The aforementioned resource allocation policies have been shown to be deterministic functions of delay QoS requirements and channel fading states. Moreover, numerical results are provided to demonstrate the performance of the proposed resource allocation policies. Copyright © 2010 John Wiley & Sons, Ltd.

Coordinated beamforming for the multicell multi-antenna wireless system

In a conventional wireless cellular system, signal processing is performed on a per-cell basis; out-of-cell interference is treated as background noise. This paper considers the benefit of coordinating base-stations across multiple cells in a multi-antenna beamforming system, where multiple base-stations may jointly optimize their respective beamformers to improve the overall system performance. Consider a multicell downlink scenario where base-stations are equipped with multiple transmit antennas employing either linear beamforming or nonlinear dirty-paper coding, and where remote users are equipped with a single antenna each, but where multiple remote users may be active simultaneously in each cell. This paper focuses on the design criteria of minimizing either the total weighted transmitted power or the maximum per-antenna power across the base-stations subject to signal-to-interference-and-noise-ratio (SINR) constraints at the remote users. The main contribution of the paper is an efficient algorithm for finding the joint globally optimal beamformers across all base-stations. The proposed algorithm is based on a generalization of uplink-downlink duality to the multicell setting using the Lagrangian duality theory. An important feature is that it naturally leads to a distributed implementation in time-division duplex (TDD) systems. Simulation results suggest that coordinating the beamforming vectors alone already provide appreciable performance improvements as compared to the conventional per-cell optimized network.

A New Perspective on Multi-User Power Control Games in Interference Channels

This paper considers the problem of how to allocate power among competing users sharing a frequency-selective interference channel. We model the interaction between selfish users as a non-cooperative game. As opposed to the existing iterative water-filling algorithm that studies the myopic users, this paper studies how a foresighted user, who knows the channel state information and response strategies of its competing users, should optimize its transmission strategy. To characterize this multi-user interaction, the Stackelberg equilibrium is introduced, and the existence of this equilibrium for the investigated noncooperative game is shown. We analyze this interaction in more detail using a simple two-user example, where the foresighted user determines its transmission strategy by solving as a bi-level program which allows him to account for the myopic user's response. It is analytically shown that a foresighted user can improve its performance, if it has the necessary information about its competitors. Since the optimal solution of Stackelberg equilibrium is computationally prohibitive, we propose a practical low-complexity approach based on Lagrangian duality theory. Numerical simulations verify the performance improvements. Possible ways to acquire the required information and to extend the formulation to more than two users are also discussed.

Distributed power and rate control for cognitive radio networks

In this paper, a distributed power and end-to-end rate control algorithm is proposed in the presence of licensed users. By Lagrangian duality theory, the optimal power and rate control solution is given for the unlicensed users while satisfying the interference temperature limits to licensed users. It is obtained that transmitting with either 0 or the maximum node power is the optimal scheme. The synchronous and asynchronous distributed algorithms are proposed to be implemented at the nodes and links. The convergence of the proposed algorithms are proved. Finally, further discussion on the utility-based fairness is provided for the proposed algorithms. Numerical results show that the proposed algorithm can limit the interference to licensed user under a predefined threshold.

Optimal distributed joint frequency, rate and power allocation in cognitive OFDMA systems

The problem of wireless resource management in broadband cognitive OFDMA networks is addressed. The objective is to maximise the multiple cognitive users' weighted rate sum by jointly adjusting their rate, frequency and power resource, under the constraints of multiple primary users' interference temperatures. First, based on two interpretations of the interference temperatures, the problem studied is formulated as two nonlinear and non-convex optimisation problems. Secondly, these two problems are analysed, and a centralised greedy algorithm is proposed to solve one problem, as well as a centralised algorithm based on Lagrangian duality theory for the other. The two centralised algorithms are shown to be optimal and both have polynomial time complexities. Finally, it is shown that the two centralized algorithms can be distributively implemented by introducing the idea of virtual clock. And the distributed algorithms can be interpreted as an interesting distributed negotiated secondary market approach. It is believed that the work will provide a good reference for the emerging cognitive network protocol design.

A simplicial branch and duality bound algorithm for the sum of convex–convex ratios problem

This article presents a simplicial branch and duality bound algorithm for globally solving the sum of convex–convex ratios problem with nonconvex feasible region. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme where the Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.

A duality-bounds algorithm for non-convex quadratic programs with additional multiplicative constraints

This article presents a duality-bounds algorithm for globally solving a non-convex quadratic programming problem (P) that contains several additional multiterm multiplicative constraints. To our knowledge, little progress has been made so far for globally solving (P). The algorithm uses a branch-and-bound scheme where Lagrange duality theory is used to obtain the lower bounds. As a result, the lower bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. Convergence of the algorithm is proved and a solved sample problem is given to show the feasibility of the proposed algorithm.

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.

A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem

In this article, we present and validate a simplicial branch and bound duality-bounds algorithm for globally solving the linear sum-of-ratios fractional program. The algorithm computes the lower bounds called for during the branch and bound search by solving ordinary linear programming problems. These problems are derived by using Lagrangian duality theory. The algorithm applies to a wide class of linear sum-of-ratios fractional programs. Two sample problems are solved, and the potential practical and computational advantages of the algorithm are indicated.

On the theory of Lagrangian duality

It is shown that a general Lagrangian duality theory for constrained extremum problems can be drawn from a separation scheme in the Image Space, namely in the space where the functions of the given problem run.

Nonsmooth model for plastic limit analysis and its smoothing algorithm

By means of Lagrange duality theory of the convex program, a dual problem of Hill’s maximum plastic work principle under Mises’ yield condition has been derived and whereby a non-differentiable convex optimization model for the limit analysis is developed. With this model, it is not necessary to linearize the yield condition and its discrete form becomes a minimization problem of the sum of Euclidean norms subject to linear constraints. Aimed at resolving the non-differentiability of Euclidean norms, a smoothing algorithm for the limit analysis of perfect-plastic continuum media is proposed. Its efficiency is demonstrated by computing the limit load factor and the collapse state for some plane stress and plain strain problems.

Decomposition Branch-and-Bound Based Algorithm for Linear Programs with Additional Multiplicative Constraints

This article presents an algorithm for globally solving a linear program (P) that contains several additional multiterm multiplicative constraints. To our knowledge, this is the first algorithm proposed to date for globally solving Problem (P). The algorithm decomposes the problem to obtain a master problem of low rank. To solve the master problem, the algorithm uses a branch-and-bound scheme where Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems in the algorithm are ordinary linear programs. Convergence of the algorithm is shown and a solved sample problem is given.

直接法による永久変形量の評価法について

Direct methods are to estimate a load factor and a displacement field at the ultimate limit state or shakedown limit state by the use of limit theorems, the shakedown theorems or other energy principles, instead of the evolutionary method which is a step by step elasto-plastic calculation in the time domain. A mathematical and mechanical structure of a direct method for the estimation of permanent deformation under the repeated loads based on the minimisation of the complementary elastic strain energy is investigated in this paper. A methodology is similar to the author's research works on the hybrid type limit / shakedown analysis alogorithms. From the point of Lagrangian duality theory, both static and kinematic analyses are derived from the proposed Lagrangian. Characteristics of the proposed method are also discussed

A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems

We study a generic minimization problem with separable nonconvex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed-integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory.