Lagrangian Duality Theory Research Articles

Augmented Lagrangian Theory, Duality and Decomposition Methods for Variational Inequality Problems

In this paper, we develop the augmented Lagrangian theory and duality theory for variational inequality problems. We propose also decomposition methods based on the augmented Lagrangian for solving complex variational inequality problems with coupling constraints.

Gradient flow approach to discrete-time envelope-constrained filter design via orthonormal filters

Using digital orthonormal filters and Lagrangian duality theory, the envelope-constrained (EC) filtering problem has been formulated as a dual quadratic programming (QP) problem with simple constraints. Applying the barrier-gradient and barrier-Newton methods based on the space transformation and gradient flow technique, two efficient design algorithms are constructed for solving this QP problem. An adaptive algorithm, based on the barrier-gradient algorithm, is developed to solve the EC filtering problem in a stochastic environment. The convergence properties are established in the mean and mean square error senses. To demonstrate the effectiveness of the proposed algorithms, a practical example using the Laguerre networks is solved for both the deterministic and stochastic cases.

Efficiency and Solution Approaches to Bicriteria Nonconvex Programs

A new nonlinear scalarization specially designed for bicriteria nonconvex programming problems is presented. The scalarization is based on generalized Lagrangian duality theory and uses an augmented Lagrange function. The new concepts, q_i-approachable points and augmented duality gap, are introduced in order to determine the location of nondominated solutions with respect to a duality gap as well as the connectedness of the nondominated set.

Controller design to optimize a composite performance measure

This paper studies a "mixed" objective problem of minimizing a composite measure of the l/sub 1/, /spl Hscr//sub 2/, and l/sub /spl infin// norms together with the l/sub /spl infin// norm of the step response of the closed loop. This performance index can be used to generate Pareto optimal solutions with respect to the individual measures. The problem is analysed for the discrete time, single-input single-output (SISO), linear time invariant systems. It is shown via the Lagrange duality theory that the problem can be reduced to a convex optimization problem with a priori known dimension. In addition, continuity of the unique optimal solution with respect to changes in the coefficients of the linear combination is established.

The singly constrained assignment problem: A Lagrangian relaxation heuristic algorithm

This manuscript presents a new heuristic algorithm to find near optimal integer solutions for the singly constrained assignment problem. The method is based on Lagrangian duality theory and involves solving a series of pure assignment problems. The software implementation of this heuristic, ASSIGN+1, successfully solved problems having one-half million binary variables (assignment arcs) in less than 17 minutes of wall clock time on a Sequent Symmetry S81 using a single processor. In computational comparisons with MPSX and OSL on an IBM 3081D, the specialized software was from 100 to 1,000 times faster. In computational comparisons with the specialized code of Mazzola and Neebe, we found that ASSIGN+1 was 40 times faster. In computational comparisons with our best alternating path specialized code, we found that ASSIGN+1 was more than three times faster than that code. This new software proved to be very robust as well as fast. The robustness is due to an elaborate scheme used to update the Lagrangean multipliers and the speed is due to the fine code used to solve the pure assignment problems. We also present a modification of the algorithm for the case in which the number of jobs exceeds the number of men along with an empirical analysis of the modified software.

Preinvexity in Multiobjective Fractional Programming

Abstract Saddle point optimality criteria and Lagrangian type duality theory are developed for a multiobjective fractional programming problem. The functions involved in the objective function and the constraints are assumed to be preinvex and not necessarily differentiable.

Quadratically constrained information theoretic analysis

Recently Zhang and Brockett extended the framework of‘minimum discrimination information’ (MDI) estimation techniques to include quadratic constraints. They claimed their approach was quite different from the usual Lagrange duality theory. We show that the dual problem obtained by Zhang and Brockett is actually a geometric dual. Hence the quadratically constrained MDI estimation can be enriched by the theory of generalized geometric programming.

A simple Duality Proof for Quadratically Constrained Entropy Functionals and Extension to Convex Constraints

In this paper, a simple proof of duality results is presented that was recently derived by Zhang and Brockett [SIAM J. Appl. Math., 47 (1987), pp. 871–885] for the problem of minimizing the Kullback–Liebler discrimination information measure subject to quadratic and linear inequality constraints. The proof is a direct and simple application of Lagrangian Duality Theory and is extended to a more general class of entropy optimization problems with composite convex constraints.

Lagrangian function and duality theory in multiobjective programming with set functions

Using the concept of vector-valued Lagrangian functions, we characterize a special class of solutions,D-solutions, of a multiobjective programming problem with set functions in which the domination structure is described by a closed convex coneD. Properties of two perturbation functions, primal map and dual map, are also studied. Results lead to a general duality theorem.

Existence and Lagrangian duality for maximizations of set-valued functions

The maximization with respect to a cone of a set-valued function into possibly infinite dimensions is defined; some existence results are established; and a Lagrangian duality theory is developed.

Optimality conditions and Lagrangian duality in continuous-time nonlinear programming

Using a perturbation approach, the Kuhn-Tucker saddlepoint and stationary-point optimality conditions and a Lagrangian duality theory are established for a general class of continuous-time nonlinear programming problems. It is shown that most of the duality formulations in the existing literature of continuous programming are special cases of this Lagrangian formulation.

Spatial and time decomposition algorithms for dynamic nonlinear network optimization using duality

The structural and computational aspects of two decomposition algorithms suitable for dynamic optimization of nonlinear interconnected networks are examined. Both methods arise from a decomposition based on Lagrangian duality theory of the addressed dynamic optimization problem, which is the minimization of energy costs over a given time period, subject to the requirement that the network equations and inequality restrictions are satisfied. The first algorithm uses a spatial decomposition of the state space into subgroups of state variables associated with particular network zones. This leads to a number of lower-dimensional optimization problems which can be solved individually at one level and coordinated at a higher level to account for interactions between these zones. The second algorithm uses time decomposition to solve a sequence of static optimization problems, one for each time increment into which the interval is subdivided, which are then coordinated to take account of dynamic interaction between the time increments. Computational results from an actual network in the United Kingdom are presented for both methods.

Geometric approaches to design of experiment

Lagrangian duality theory is used to relate problems of D-optimal design of experiments to covering problems involving ellipsoids and ellipsoidal cylinders. The possibility of generalisation of the results to other optimality criteria is discussed. The covering ellipsoids find application in some robust estimation activities and there are links with the operational research problems of optimal location of facilities. The possibility of exact computation of the covering ellipsoids is investigated.

Reliability‐constrained dynamic programing and randomized release rules in reservoir management

A stochastic dynamic programing model for determining reservoir release rules which maximize expected net benefits subject to reliability constraints on system performance is presented within the framework of Lagrangian duality theory. The approach characterizes optimal release rules under discounting of benefits as being nonstationary. It also indicates when there exists a randomized rule which dominates the optimal deterministic rule. The methodology for obtaining these rules uses dual optimization to provide bounds on the optimal solution and a branch and bound procedure to resolve any duality gap which may occur if the bounds are not equal. The model is applied to a group of hypothetical problems for which nonoptimal policies had previously been found by using chance‐constrained dynamic programing (Askew, 1974b).

Decentralized optimization for distributed-lag models of discrete systems

The approach discussed in this paper solves a general class of optimization problems for discrete dynamic systems which include distributed lag, distributed and/or multiple pure delays, and constraints both in state and control variables. The overall system equation of this problem is described by a multidimensional nonlinear difference equation of high-order which is called the distributed-lag model. Applying Lagrange duality theory to the original problem, the dual problem is formulated, and the decomposition of the optimization process in stage is obtained. It is shown that by solving the dual problem the delay terms can be easily handled and the optimal solution to the original problem is obtained without reducing the multi-dimensional high-order system equation to a conventional larger dimensional first-order system equation. It is also shown that the dual decentralized method in this paper is easier to cope with state and control constraints than the primal method in the space of control, i.e. gradient and other techniques. The approach developed in this paper is compared with other methods using a simple example, and is applied to a combined marketing and production control problem. Some computational results are included.

Hierarchical optimisation of a water-supply network

The paper presents the results of an investigation of the feasibility and implementation of an optimal online control to a water-supply network in the United Kingdom. The network, which has over 70 principal nodes, is reduced to an equivalent control problem with 6 state variables, which are the dominant reservoir heights, 10 control inputs, which are the major pump-station outputs, and 6 disturbances, which are the lumped zone consumptions. The control objective is the supply of water to the consumer over a 24 h period at minimum cost, subject to the 16 constraints on controls and states at each interval. To overcome the dimensionality problem associated with discrete-time optimal control the optimisation is carried out using a decomposition technique due to Lasdon and Tamura, which employs Lagrange duality theory. The paper describes the formulation of the network equations, the theory of the optimisation method, and gives typical results for the optimal operation of the system for typical weekday consumption. The results show a worthwhile cost saving over the control strategy in use during the study period, and computational experience indicates that the method is appropriate for optimal online control in a predictive environment.

Hierarchical optimisation of a water-supply network

The paper presents the results of an investigation of the feasibility and implementation of an optimal online control to a water-supply network in the United Kingdom. The network, which has over 70 principal nodes, is reduced to an equivalent control problem with 6 state variables, which are the dominant reservoir heights, 10 control inputs, which are the major pump-station outputs, and 6 disturbances, which are the lumped zone consumptions. The control objective is the supply of water to the consumer over a 24 h period at minimum cost, subject to the 16 constraints on controls and states at each interval. To overcome the dimensionality problem associated with discrete-time optimal control the optimisation is carried out using a decomposition technique due to Lasdon and Tamura, which employs Lagrange duality theory. The paper describes the formulation of the network equations, the theory of the optimisation method, and gives typical results for the optimal operation of the system for typical weekday consumption. The results show a worthwhile cost saving over the control strategy in use during the study period, and computational experience indicates that the method is appropriate for optimal online control in a predictive environment.