A polynomially bounded algorithm for a singly constrained quadratic program

A survey on the continuous nonlinear resource allocation problem

“…The first (recursive) pegging algorithm [San71] The first article to discuss both pegging and Lagrange multiplier algorithms [LuG75] The first article to discuss the value of having an explicit formula for (µ) [BiH77] The first true pegging algorithm, together with convergence theory [HKL80] The first complexity analysis of a Lagrange multiplier algorithm [Zip80b] The first survey on algorithms [Zip80b] The first discussion on the reoptimization of the problem for small changes in the data; utilizes the previous value of µ * [Ein81]…”

“…[Lot06] P. A. Lotito, Issues in the implementation of the DSD algorithm for the traffic assignment problem (Problem) φj(xj) = q j 2 x 2 j − rjxj, qj ≥ 0; linear equality (aj = 1, b = 1); lj = 0, uj = ∞ (Origin) Subproblem for each origin-destination pair in the traffic assignment problem within the scaled reduced gradient method of [LaP92] (Methodology) A Newton method, wherein q , at breakpoints where it is not defined, is replaced by the left (right) derivative of q when q is negative (positive) (Citations) Refers to [HKL80,Bru84,PaK90] for the case when qj > 0 for all j, and to [NiZ92] for a similar Newton method (Notes) Numerical experiments on median search, randomized median search and the proposed Newton method (n ∈ [100, 400]); they show similar performance and complexity, but the Newton method is slightly faster. Has observed in actual iterative use for the traffic assignment problem that the latter is even faster.…”

A Class of Mathematical Programs with Equilibrium Constraints: A Smooth Algorithm and Applications to Contact Problems

“…In this case the nonlinear program (22) is replaced by a strictly convex quadratic knapsack problem, which can be processed by a number of quite efficient polynomial algorithms (see Helgason et al, 1980;Pardalos and Kovoor, 1990;Robinson et al, 1992). (ii) Computation of f (x) and ∇ x f (x) and implementation of the Armijo Criterion-It follows from (10) and (12) that, for eachx ∈ X ,…”

“…The equation E = n j=1 [E f x j + E m (1 − x j )] is a linear constraint that should be included in the definition of the set X . The projected-gradient algorithm can also be applied in this case, but the projection operator P X should be computed by one of the algorithms described in Helgason et al (1980) and Robinson et al (1992); Pardalos and Kovoor (1990), for this so-called strictly convex quadratic knapsack problem.…”

A breakpoint search approach for convex resource allocation problems with bounded variables

“…The idea of the breakpoint search approach then is to identify two consecutive breakpoints where the function g(·) has opposite sign. Then, g(λ) is linear between these two breakpoints, and the optimal Lagrange multiplier is found through linear interpolation (see, e.g., [7,8,11,12,16]). We generalize this approach to convex resource allocation problems, with specific emphasis on how the Lagrange multiplier is found once the breakpoint search is completed, and how the breakpoint search yields information on whether or not bounds are binding.…”