Multiplier Vector Research Articles

Distribution System Restoration via Subgradient-Based Lagrangian Relaxation

Lagrangian relaxation with subgradient iterations is applied to the problem of optimal restoration of distribution systems after a blackout. The method breaks the restoration horizon into intervals and develops a restoration plan by finding the status of each of the feeders at each time interval. The Lagrangian relaxation approach allows a computationally efficient calculation of the timing and selection of feeders to be energized. The subgradient iterative approach relates to the selection of a vector of Lagrange multipliers in the optimization process. A restoration index obtained from the dual formulation of the problem is also shown. This index identifies the feeders that are closer to restoration. The proposed algorithm applies to radially configured distribution systems and is an operator-permissive, automated approach. Examples are shown including a 100-feeder restoration case.

Asymptotic analysis of the trajectories of the logarithmic barrier algorithm without constraint qualifications

In this paper, we study the differentiability of the trajectories of the logarithmic barrier algorithm for a nonlinear program when the set A* of the Karush-Kuhn-Tucker multiplier vectors is empty owing to the fact that the constraint qualifications are not satisfied.

SU‐FF‐T‐91: Automated Selection of Tradeoff Parameters in IMRT Prioritized Prescription Treatment Planning

Purpose: Prioritized prescription optimization is a method of IMRT treatment planning which solves the optimization problem step‐wise, first by determining the best performance of a high‐priority objective and then converting it to a constraint as lower priority goals are optimized in turn. This system is fully automated; however, there is a variable value we call ‘slip’ factor which determines the amount of degradation allowed after each priority is addressed. The purpose of this work is to investigate an approach to automating selection of the slip factor. Method and Materials: We propose a new version of prioritized prescription optimization that, after each step, estimates the desired value of the slip factor for the next step. We estimate, to first order, the rate of change of the objective function with respect to small perturbations in the slip. We then apply the Sensitivity Theorem to analyze the effect of the slip factor. In summary, after solving the first problem and obtaining the solution to the first step, wI, we plug the value of wI into a closed‐form expression of the Lagrange multiplier vector λ(0) to get λ(0), and then use the last element of λ(0) to compute the first‐order approximation model for the optimal objective value with variable slip. Results: We use the Sensitivity Theorem and Lagrange multipliers to derive a first‐order approximation model for the optimal objective function value for a given slip factor. On this basis, we propose to automatically select slip values based on the expected gain in the objective function. Conclusion: We have developed a mathematical theory and framework which allows us to automate the selection of the slip factor in prioritized prescription optimization, based on a local linear approximation. This proposal is the subject of ongoing tests.

On the stability of a convex set of matrices

In this paper we give an alternative proof of the constant inertia theorem for convex compact sets of complex matrices. It is shown that the companion matrix whose non-trivial column is negative satisfies the directional Lyapunov condition (inclusion) for real multiplier vectors. An example of a real matrix polytope that satisfies the directional Lyapunov condition for real multiplier vectors and which has nonconstant inertia is given. A new stability criterion for convex compact sets of real Z-matrices is given. This criterion uses only real vectors and positive definite diagonal matrices.

A Scatter Search Method for the Bi-Criteria Multi-dimensional {0,1}-Knapsack Problem using Surrogate Relaxation

This paper presents a scatter search (SS) based method for the bi-criteria multi-dimensional knapsack problem. The method is organized according to the usual structure of SS: (1) diversification, (2) improvement, (3) reference set update, (4) subset generation, and (5) solution combination. Surrogate relaxation is used to convert the multi-constraint problem into a single constraint one, which is used in the diversification method and to evaluate the quality of the solutions. The definition of the appropriate surrogate multiplier vector is also discussed. Tests on several sets of large size instances show that the results are of high quality and an accurate description of the entire set of the non-dominated solutions can be obtained within reasonable computational time. Comparisons with other meta-heuristics are also presented. In the tested instances the obtained set of potentially non-dominated solutions dominates the set found with those meta-heuristics.

Idempotence and Orthogonality in Relation to Mixed Model Adjustments

The purpose of this paper is to look into several common idempotent matrices of interest in the context of any mixed model least-squares adjustment. The cofactor (scaled covariance) matrices, based on the law of error propagation, are incorporated into the least-squares estimation solutions. The rank of an idempotent matrix is given by taking the matrix trace, which leads to an unbiased estimate of the unit weight variance. The orthogonality between two projected vector subspaces is proven by using the generalized inner products, or equivalently, according to the cosine law. The Lagrange multiplier vector and its associated cofactor matrix are then employed to verify the already determined idempotent as well as orthogonal properties, reflecting the correctness of the formulated matrix equations.

Some remarks on multiplier ideals and vector bundles

In this paper, we give two applications of the theory of multiplier ideals to vector bundles over complex projective manifolds, generalizing to higher rank results already established for line bundles. The first addresses the existence of sections of (suitable twists) of symmetric powers of a very ample vector bundle, vanishing on a given subvariety. The second is a vanishing theorem of Gri‰ths type, adjusted with an additional multiplier ideal term as in the standard Nadel vanishing theorem. To begin with, we recall that, starting with work of Bombieri and Skoda, there has been considerable interest in going from hypersurfaces in P highly singular at a given set of points S to hypersurfaces through S of relatively low degree; there is now a very direct and terse approach based on multiplier ideals [3], [4], [5], [6]. Here, the same method is shown to yield a statement in the same spirit about sections of very ample vector bundles on a projective manifold. One says that a rank r holomorphic vector bundle on a complex projective manifold is very ample if the relative hyperplane line bundle OPE ð1Þ on the projectivised dual PE is very ample. If X is a projective manifold and ZJX an irreducible subvariety, for every integer pd 1 the p-th symbolic power I hpi Z JOZ of the ideal sheaf of Z is the ideal sheaf of the holomorphic functions vanishing with multiplicity dp along Z (that is, at a generic point of Z ).

AN IMPROVED SURROGATE CONSTRAINTS METHOD FOR SEPARABLE NONLINEAR INTEGER PROGRAMMING

An improved surrogate constraints method for solving separable nonlinear integer programming problems with multiple constraints is presented. The surrogate constraints method is very effective in solving problems with multiple constraints. The method solves a succession of surrogate constraints problems having a single constraint instead of the original multiple constraint problem. A surrogate problem with an optimal multiplier vector solves the original problem exactly if there is no duality gap. However, the surrogate constraints method often has a duality gap, that is it fails to find an exact solution to the original problem. The modification proposed closes the surrogate duality gap. The modification solves a succession of target problems that enumerates all solutions hitting a particular target. The target problems are produced by using an optimal surrogate multiplier vector. The computational results show that the modification is very effective at closing the surrogate gap of multiple constraint problems.

Efficient Aerodynamic Design Method Using a Tightly Coupled Algorithm

An efe cient aerodynamic design optimization method for the compressible Euler equations is presented. A gradient-based optimization method is used to e nd the optimal design, in conjunction with a continuous adjoint sensitivity analysis method. To improve the efe ciency of the design method, a tightly coupled algorithm based on a step-size estimation isproposed. The efe ciency of thepresentmethod is demonstrated through drag minimizations of a wing and an airfoil. The result shows that the cost of the present method is signie cantly lower than that of the loosely coupled algorithm. Nomenclature Ci = e ux Jacobian matrix in the computational domain d = design variables vector Fi = e ux vector in the computational domain fi = e ux vector in the physical domain G = gradient vector of the object function I = object function q = conservative e ow variables vector R = residual vector in the computational domain S = searching direction ≤ = step size along the searching direction S A = Lagrangian multiplier vector Subscript face = value at cell face Superscript T = transpose of vector or matrix

A multiparametric programming approach for mixed-integer quadratic engineering problems

In this work we propose algorithms for the solution of multiparametric quadratic programming (mp-QP) problems and multiparametric mixed-integer quadratic programming (mp-MIQP) problems with a convex and quadratic objective function and linear constraints. For mp-QP problems it is shown that the optimal solution, i.e. the vector of continuous variables and Lagrange multipliers, is an affine function of parameters. The basic idea of the algorithm is to use this affine expression for the optimal solution to systematically characterize the space of parameters by a set of regions of optimality. The solution of the mp-MIQP problems is approached by decomposing it into two subproblems, which converge based upon an iterative methodology. The first subproblem, which is an mp-QP, is obtained by fixing the integer variables and its solution represents a parametric upper bound. The second subproblem is formulated as a mixed-integer non-linear programming (MINLP) problem and its solution provides a new integer vector, which can be fixed to obtain a parametric solution, which is better than the current upper bound. The algorithm terminates with an envelope of parametric profiles corresponding to different optimal integer solutions. Examples are presented to illustrate the basic ideas of the algorithms and their application in model predictive and hybrid control problems.

A Constrained Multibody System Dynamics Avoiding Kinematic Singularities.

In the analysis of constrained multibody systems, the constraint reaction forces are normally expressed in terms of the constraint equations and a vector of Lagrange multipliers. Because it fails to incorporate conservation of momentum, the Lagrange multiplier method is deficient when the constraint Jacobian matrix is singular. This paper presents an improved dynamic formulation for the constrained multibody system. In our formulation, the kinematic constraints are still formulated in terms of the joint constraint reaction forces and moments; however, the formulations are based on a second-order Taylor expansion so as to incorporate the rigid body velocities. Conservation of momentum is included explicitly in this method; hence the problems caused by kinematic singularities can be avoided. In addition, the dynamic formulation is general and applicable to most dynamic analyses. Finally the 3-leg Stewart platform is used for the example of analysis.

Time precise integration method for constrained nonlinear control system

For the constrained nonlinear optimal control problem, by taking the first term of Taylor series, the dynamic equation is linearized. Thus by introducing into the dual variable (Langrange multiplier vector), the dynamic equation can be transformed into Hamilton system from Lagrange system on the basis of the original variable. Under the whole state, the problem discussed can be described from a new view, and the equation can be precisely solved by the time precise integration method established in linear dynamic system. A numerical example shows the effectiveness of the method.

A full step Newton method for the solution of a nonlinear magnetostatic optimization problem

We consider a nonlinear magnetostatic field problem, where in a certain subdomain a cost functional (i.e., the magnetic field has to fulfill a prescribed figure) is defined. The optimization problem is stated in such a way that the field problem is treated as a constraint. A Lagrange function is established depending on the magnetic vector potential A, the current density J in coil segments and the vector of Lagrange multipliers /spl lambda/. The derivatives with respect to the variables are set to zero to obtain the optimality system. The optimality system is solved applying full Newton steps without line search. For the left hand side of the Newton system the second derivatives of the Lagrange function are required while the first derivatives of the Lagrange function are written into the right hand side. Employing a finite element scheme to discretize the problem domain, we end up with a linear system of equations that has to be solved for each Newton step.

Scoring with constraints

AbstractThis paper considers the solution of estimation problems based on the maximum likelihood principle when a fixed number of equality constraints are imposed on the parameters of the problem. Consistency and the asymptotic distribution of the parameter estimates are discussed as n → ∞, where n is the number of independent observations, and it is shown that a suitably scaled limiting multiplier vector is known. It is also shown that when this information is available then the good properties of Fisher's method of scoring for the unconstrained case extend to a class of augmented Lagrangian methods for the constrained case. This point is illustrated by means of an example involving the estimation of a mixture density.

Coevolutionary augmented Lagrangian methods for constrained optimization

This paper introduces a coevolutionary method developed for solving constrained optimization problems. This algorithm is based on the evolution of two populations with opposite objectives to solve saddle-point problems. The augmented Lagrangian approach is taken to transform a constrained optimization problem to a zero-sum game with the saddle point solution. The populations of the parameter vector and the multiplier vector approximate the zero-sum game by a static matrix game, in which the fitness of individuals is determined according to the security strategy of each population group. Selection, recombination, and mutation are done by using the evolutionary mechanism of conventional evolutionary algorithms such as evolution strategies, evolutionary programming, and genetic algorithms. Four benchmark problems are solved to demonstrate that the proposed coevolutionary method provides consistent solutions with better numerical accuracy than other evolutionary methods.

Pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.

Modified Wilson's Method for Nonlinear Programs with Nonunique Multipliers

This paper first investigates Newton-type methods for generalized equations with compact solution sets. The analysis of their local convergence behavior is based, besides other conditions, on the upper Lipschitz-continuity of the local solution set mapping of a simply perturbed generalized equation. This approach is then applied to the KKT conditions of a nonlinear program with inequality constraints and leads to a modified version of the classical Wilson method. It is shown that the distances of the iterates to the set of KKT points converge q-quadratically to zero under conditions that do not imply a unique multiplier vector. Additionally to the Mangasarian-Fromovitz Constraint Qualification and to a Second-Order Sufficiency Condition the local minimizer is required to fulfill a Constant Rank Condition (weaker than the Constant Rank Constraint Qualification one of Janin) and a so-called Weak Complementarity Condition.

Lagrangean/surrogate relaxation for generalized assignment problems

This work presents Lagrangean/surrogate relaxation to the problem of maximum profit assignment of n tasks to m agents ( n > m), such that each task is assigned to only one agent subject to capacity constraints on the agents. The Lagrangean/surrogate relaxation combines usual Lagrangean and surrogate relaxations relaxing first a set of constraints in the surrogate way. Then, the Lagrangean relaxation of the surrogate constraint is obtained and approximately optimized (one-dimensional dual). The Lagrangean/surrogate is compared with the usual Lagrangean relaxation on a computational study using a large set of instances. The dual bounds are the same for both relaxations, but the Lagrangean/surrogate can give improved local bounds at the application of a subgradient method, resulting in less computational times. Three relaxations are derived for the problem. The first relaxation considers a vector of multipliers for the capacity constraints, the second for the assignment constraints and the other for the Lagrangean decomposition constraints. Relaxation multipliers are used with efficient constructive heuristics to find good feasible solutions. The application of a Lagrangean/surrogate approach seems very promising for large scale problems.

What Is the Economic Meaning of FDH

The central feature of the FDH model is the lack of convexity for its production possibility set, TF. Starting with n observed (distinct) decision making units DMUk , each defined by an input-output vector p k = [y k -x k], domination is defined by ordinary vector inequalities. DMUk is said to dominate DMUj if p k ≥ p j , p k ≠ p j . The FDH production possibility set TF consists of the observed DMUj together with all input-output vectors p=[yk,−xk] with y ≥ 0, x ≥ 0, y ≠ 0, x ≠ 0 which are dominated by at least one of the observed DMUj. DMUk is defined as “FDH efficient” if no DMUj dominates it. In the BCC (or variable return to scale) DEA model the production possibility set TB consists of the observed DMUk together with all input-output vectors dominated by any convex combination of them and DMUk is DEA efficient if it is not dominated by any p in TB. In the DEA model, economic meaning is established by the introduction of (non negative) multiplier (price) vectors w = [u,v]. If DMUk is undominated (in TB) then there exists a positive multiplier vector w for which (a) w T p k = u T y k − v T x k ≥ w T p for every p ∈ TB. In everyday language, the net return (or profit) for DMUk relative to the given multiplier vector w is at least as great as that for any production possibility p. On the other hand, if DMUk is FDH but not DEA efficient then it is proved that there exists no positive multiplier vector >w for which (a) holds, i.e. for any positive w there exists at least one DMUj for which w T p j > w T p k . Since, therefore, FDH efficiency does not guarantee price efficiency what is its economic significance? Without economic significance, how can FDH be considered as being more than a mathematical system however logically soundly it may be conceived?

Unit commitment with ramp multipliers

This paper presents a new decomposition method, based on the Lagrangian relaxation technique, for solving the unit commitment problem with ramp rate constraints. By introducing an additional vector of multipliers to represent the cost of "system ramping demand", this method can handle the coupling constraints between time periods while still keeping the simplicity of the original decomposition method. A new algorithm for updating multipliers is also proposed. Similar to the bundle algorithm, this algorithm maintains the previous iteration history to approximate the dual envelope. Unlike the bundle algorithm, this new algorithm generates an update step along the subgradient direction without any quadratic programming (QP) code. The new algorithm combines the bundle algorithm's smooth approach to the dual optimum with the sub-gradient method's fast update.