Dual Optimal Solutions Research Articles

Strong and compact relaxations in the original space using a compact extended formulation

For certain integer programs, one way to obtain a strong dual bound is to use an extended formulation and then solve the associated linear programming relaxation.The classical way to obtain a bound of the same value in the original variable space is through the use of Benders’ algorithm. Here, we propose an alternative approach based on a decomposition of the dual optimal solution of the extended formulation linear program. An example of the approach using the multi-commodity formulation of a two-level production/transportation problem is presented.

A New Method for Deriving Robust and Globalized Robust Solutions of Uncertain Linear Conic Optimization Problems Having General Convex Uncertainty Sets

We propose a new way to derive tractable robust counterparts of a linear conic optimization problem by using the theory of Beck and Ben-Tal on the duality between the robust (“pessimistic”) primal problem and its “optimistic” dual. First, we obtain a new convex reformulation of the dual problem of a robust linear conic program, and then show how to construct the primal robust solution from the dual optimal solution. Our result allows many new uncertainty regions to be considered. We give examples of tractable uncertainty regions that were previously intractable. The results are illustrated by solving a multi-item newsvendor problem. We also propose a new globalized robust counterpart that is more flexible, and is tractable for general convex uncertainty sets and any convex distance function.

Distributed primal–dual stochastic subgradient algorithms for multi‐agent optimization under inequality constraints

SUMMARYWe consider the multi‐agent optimization problem where multiple agents try to cooperatively optimize the sum of their local convex objective functions, subject to global inequality constraints and a convex constraint set over a network. Through characterizing the primal and dual optimal solutions as the saddle points of the associated Lagrangian function, which can be evaluated with stochastic errors, we propose the distributed primal–dual stochastic subgradient algorithms for two cases: (i) the time model is synchronous and (ii) the time model is asynchronous. In the first case, we obtain bounds on the convergence properties of the algorithm for a diminishing step size. In the second case, for a constant step size, we establish some error bounds on the algorithm's performance. In particular, we prove that the error bounds scale as in the number of n agents. Copyright © 2012 John Wiley & Sons, Ltd.

Low-complexity optimal discrete-rate spectrum balancing in digital subscriber lines

Discrete-rate spectrum balancing in interference-limited multi-user and multi-carrier digital subscriber lines (DSL) is a large-scale, non-convex and combinatorial problem. Previously proposed algorithms for its (dual) optimal solution are only applicable for networks with few users, while the suboptimality of less complex bit-loading algorithms has not been adequately studied so far. We deploy constrained optimization techniques as well as problem-specific branch-and-bound and search-space reduction methods, which for the first time give a low-complexity guarantee of optimality in certain multi-user DSL networks of practical size. Simulation results precisely quantify the suboptimality of multi-user bit-loading schemes in a thousand ADSL2 scenarios under measured channel data.

Distributed Primal–Dual Subgradient Method for Multiagent Optimization via Consensus Algorithms

This paper studies the problem of optimizing the sum of multiple agents' local convex objective functions, subject to global convex inequality constraints and a convex state constraint set over a network. Through characterizing the primal and dual optimal solutions as the saddle points of the Lagrangian function associated with the problem, we propose a distributed algorithm, named the distributed primal-dual subgradient method, to provide approximate saddle points of the Lagrangian function, based on the distributed average consensus algorithms. Under Slater's condition, we obtain bounds on the convergence properties of the proposed method for a constant step size. Simulation examples are provided to demonstrate the effectiveness of the proposed method.

Delay reduction via Lagrange multipliers in stochastic network optimization

In this paper, we consider the problem of reducing network delay in stochastic network utility optimization problems. We start by studying the recently proposed quadratic Lyapunov function based algorithms (QLA, also known as the MaxWeight algorithm). We show that for every stochastic problem, there is a corresponding deterministic problem, whose dual optimal solution “exponentially attracts” the network backlog process under QLA. In particular, the probability that the backlog vector under QLA deviates from the attractor is exponentially decreasing in their Euclidean distance. This is the first such result for the class of algorithms built upon quadratic Lyapunov functions. The result quantifies the “network gravity” role of Lagrange Multipliers in network scheduling. It not only helps to explain how QLA achieves the desired performance but also suggests that one can roughly “subtract out” a Lagrange multiplier from the system induced by QLA.

An interior–exterior approach for convex quadratic programming

In the last decade, a new class of interior–exterior algorithms for linear programming was developed. The method was based on the use of mixed penalty function with two separate parameters to solve a set of sub-penalized problems associated to the initial problem.To study the necessary optimality conditions, one introduced a new concept of the so-called pseudo-gap to describe fully the optimal primal and dual solutions. Only one Newton iteration is sufficient to approximate the solution of penalized problem which satisfies a criterion of proximity.The purpose of this work is to extend the approach to the convex quadratic programming problems.

Min-cost multiflows in node-capacitated undirected networks

We consider an undirected graph G=(VG,EG) with a set T⊆VG of terminals, and with nonnegative integer capacities c(v) and costs a(v) of nodes v∈VG. A path in G is a T-path if its ends are distinct terminals. By a multiflow we mean a function F assigning to each T-path P a nonnegative rational weight F(P), and a multiflow is called feasible if the sum of weights of T-paths through each node v does not exceed c(v). The value of F is the sum of weights F(P), and the cost of F is the sum of F(P) times the cost of P w.r.t. a, over all T-paths P. Generalizing known results on edge-capacitated multiflows, we show that the problem of finding a minimum cost multiflow among the feasible multiflows of maximum possible value admits half-integer optimal primal and dual solutions. Moreover, we devise a strongly polynomial algorithm for finding such optimal solutions.

A ranking model for the greedy algorithm and discrete convexity

Generalizing the idea of the Lovász extension of a set function and the discrete Choquet integral, we introduce a combinatorial model that allows us to define and analyze matroid-type greedy algorithms. The model is based on a real-valued function v on a (finite) family of sets which yields the constraints of a combinatorial linear program. Moreover, v gives rise to a ranking and selection procedure for the elements of the ground set N and thus implies a greedy algorithm for the linear program. It is proved that the greedy algorithm is guaranteed to produce primal and dual optimal solutions if and only if an associated functional on $${\mathbb{R}^N}$$ is concave. Previous matroid-type greedy models are shown to fit into the present general context. In particular, a general model for combinatorial optimization under supermodular constraints is presented which guarantees the greedy algorithm to work.

Accelerating the Benders decomposition for network-constrained unit commitment problems

This paper presents an optimization method by generating multiple strong Benders cuts for accelerating the convergence of Benders Decomposition (BD) when solving the network-constrained generation unit commitment (NCUC) problem. In NCUC, dc transmission network evaluation subproblems are highly degenerate, which would lead to many dual optimal solutions. Furthermore, the classical BD cuts are often low-density which involve only a limited number of decision variables in the master problem. Therefore, the dual optimal solutions and the corresponding Benders cuts are of crucial importance for improving the efficiency of the BD algorithm. The proposed method would generate multiple strong Benders cuts, which are pareto optimal, among candidates from multiple dual optimal solutions. Such cuts would be high-density in comparison with low-density cuts produced by the classical BD. The proposed multiple strong Benders cuts are efficient in terms of reducing the total iteration number and the overall computing time. The high-density cuts may restrict the feasible region of the master unit commitment (UC) problem in each iteration as they incorporate more decision variables in each Benders cut. The multiple strong Benders cuts would accordingly reduce the iteration number and overall computing time. Numerical tests demonstrate the efficiency of the proposed multiple strong Benders cuts method in comparison with the classical BD algorithm and the linear sensitivity factors (LSF) method. The proposed method can be extended to other applications of BD for solving the large-scale optimization problems in power systems operation, maintenance, and planning.

Reduced Basis Method and A Posteriori Error Estimation for Parametrized Linear-Quadratic Optimal Control Problems

We propose the reduced basis method for the solution of parametrized optimal control problems described by parabolic partial differential equations in the unconstrained case. The method, which is based on an off-line-on-line decomposition procedure, allows at the on-line step large computational cost savings with respect to the “truth” approximation used for defining the reduced basis. An a posteriori error estimate is provided by means of the goal-oriented analysis, thus associating an error bound to each optimal solution of the parametrized optimal control problem and answering to the demand for a reliable method. An adaptive procedure, led by the a posteriori error estimate, is considered for the generation of the reduced basis space, which is set according to the optimal primal and dual solutions of the optimal control problem at hand.

An exact solution framework for a broad class of vehicle routing problems

This paper presents an exact solution framework for solving some variants of the vehicle routing problem (VRP) that can be modeled as set partitioning (SP) problems with additional constraints. The method consists in combining different dual ascent procedures to find a near optimal dual solution of the SP model. Then, a column-and-cut generation algorithm attempts to close the integrality gap left by the dual ascent procedures by adding valid inequalities to the SP formulation. The final dual solution is used to generate a reduced problem containing all optimal integer solutions that is solved by an integer programming solver. In this paper, we describe how this solution framework can be extended to solve different variants of the VRP by tailoring the different bounding procedures to deal with the constraints of the specific variant. We describe how this solution framework has been recently used to derive exact algorithms for a broad class of VRPs such as the capacitated VRP, the VRP with time windows, the pickup and delivery problem with time windows, all types of heterogeneous VRP including the multi depot VRP, and the period VRP. The computational results show that the exact algorithm derived for each of these VRP variants outperforms all other exact methods published so far and can solve several test instances that were previously unsolved.

Investment, uncertainty, and production games

ABSTRACTThis paper explores a few cooperative aspects of investments in uncertain, real options. By hypothesis some production commitments, factors, or quotas are transferable. Cases in point include energy supply, emission of pollutants, and harvest of renewable resources. Of particular interest are technologies or projects that provide anti-correlated returns. Any such project stabilizes the aggregate proceeds. Therefore, given widespread risk aversion, a project of this sort merits a bonus. The setting is formalized as a two-stage, stochastic, production game. Absent economies of scale, such games are quite tractable in analysis, computation, and realization. A core imputation comes in terms of shadow prices that equilibrate competitive, endogenous markets. Such prices emerge as optimal dual solutions to coordinated production programs, featuring pooled commitments, or resources. Alternatively, the prices could result from repeated exchange.

Generating and measuring instances of hard semidefinite programs

Linear Programming, LP, problems with finite optimal value have a zero duality gap and a primal–dual strictly complementary optimal solution pair. On the other hand, there exist Semidefinite Programming, SDP, problems which have a nonzero duality gap (different primal and dual optimal values; not both infinite). The duality gap is assured to be zero if a constraint qualification, e.g., Slater’s condition (strict feasibility) holds. Measures of strict feasibility, also called distance to infeasibility, have been used in complexity analysis, and, it is known that (near) loss of strict feasibility results in numerical difficulties. In addition, there exist SDP problems which have a zero duality gap but no strict complementary primal–dual optimal solution. We refer to these problems as hard instances of SDP. The assumption of strict complementarity is essential for asymptotic superlinear and quadratic rate convergence proofs. In this paper, we introduce a procedure for generating hard instances of SDP with a specified complementarity nullity (the dimension of the common nullspace of primal–dual optimal pairs). We then show, empirically, that the complementarity nullity correlates well with the observed local convergence rate and the number of iterations required to obtain optimal solutions to a specified accuracy, i.e., we show, even when Slater’s condition holds, that the loss of strict complementarity results in numerical difficulties. We include two new measures of hardness that correlate well with the complementarity nullity.

Total Dual Integrality of Rothblum's Description of the Stable-Marriage Polyhedron

Rothblum showed that the convex hull of the stable matchings of a bipartite preference system can be described by an elegant system of linear inequalities. In this paper we prove that the description given by Rothblum is totally dual integral. We give a constructive proof based on the results of Gusfield and Irving on rotations, which gives rise to a strongly polynomial algorithm for finding an integer optimal dual solution.

Dual periodicity in [formula omitted]-norm minimisation problems

The topic of this paper is the discrete-time l 1 -norm minimisation problem with convolution constraints. We find primal initial conditions for which the dual optimal solution is periodic. Periodicity of the dual optimal solution implies satisfaction of a simple linear recurrence relation by the primal optimal solution.

Sensitivity Analysis in Calculus of Variations. Some Applications

This paper deals with the problem of sensitivity analysis in calculus of variations. A perturbation technique is applied to derive the boundary value problem and the system of equations that allow us to obtain the partial derivatives (sensitivities) of the objective function value and the primal and dual optimal solutions with respect to all parameters. Two examples of applications, a simple mathematical problem and a slope stability analysis problem, are used to illustrate the proposed method.

Limiting behavior of the Alizadeh–Haeberly–Overton weighted paths in semidefinite programming

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming obtained from centrality equations of the form X S+SX=2 ν W, where W is a fixed positive definite matrix and ν>0 is a parameter, under the assumption that the problem has a strictly complementary primal–dual optimal solution. We present a different and simpler proof than the one given by Preiß and Stoer [Preiß, M. and Stoer, J., 2004, Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems. Mathematical Programming 99, 499–520.] that a weighted central path as a function of ν can be extended analytically beyond 0. In addition, the characterization of the limit points of the path and its normalized first-order derivatives is also provided. We also derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primal–dual optimal solutions. Finally, we make some observation for the superlinear convergence of some primal–dual interior point SDP algorithms using AHO neighborhood.

An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts

This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved by an integer programming solver. Computational results over the main instances from the literature show the effectiveness of the proposed algorithm.

Active Constraint Set Invariancy Sensitivity Analysis in Linear Optimization

Active constraint set invariancy sensitivity analysis is concerned with finding the range of parameter variation so that the perturbed problem has still an optimal solution with the same support set that the given optimal solution of the unperturbed problem has. However, in an optimization problem with inequality constraints, active constraint set invariancy sensitivity analysis aims to find the range of parameter variation, where the active constraints in a given optimal solution remains invariant. For the sake of simplicity, we consider the primal problem in standard form and consequently its dual may have an optimal solution with some active constraints. In this paper, the following question is answered: "what is the range of the parameter, where for each parameter value in this range, a dual optimal solution exists with exactly the same set of positive slack variables as for the current dual optimal solution?". The differences of the results between the linear and convex quadratic optimization problems are highlighted too.