Dual LP Research Articles

Depth-3 circuits for inner product

What is the Σ32-circuit complexity (depth 3, bottom-fanin 2) of the 2n-bit inner product function? The complexity is known to be exponential 2αnn for some αn=Ω(1). We show that the limiting constant α≔limsupαn satisfies0.847...≤α≤0.965.... Determining α is one of the seemingly-simplest open problems about depth-3 circuits. The question was recently raised by Golovnev, Kulikov, and Williams (ITCS 2021) and Frankl, Gryaznov, and Talebanfard (ITCS 2022), who observed that α∈[0.5,1]. To obtain our improved bounds, we analyse a covering LP that captures the Σ32-complexity up to polynomial factors. In particular, our lower bound is proved by constructing a feasible solution to the dual LP.

Lp Dual Mixed Affine Quermassintegrals

Drawn-out dual Lp-Brunn-Minkowski inequality, some ordinary and useful characters and the uniqueness in the matter of Lp dual mixed affine quermassintegrals are discussed, and the equivalence of the inequality concerning Lp harmonic combinations, and its dual Lp- Minkowski inequality is proved.

SIR Epidemics with State-Dependent Costs and ICU Constraints: A Hamilton–Jacobi Verification Argument and Dual LP Algorithms

The aim of this paper is twofold. On one hand, we strive to give a simpler proof of the optimality of greedy controls when the cost of interventions is control-affine and the dynamics follow a state-constrained controlled SIR model. This is achieved using the Hamilton–Jacobi characterization of the value function, via the verification argument and explicit trajectory-based computations. Aside from providing an alternative to the Pontryagin complex arguments in Avram et al. (Appl Math Comput 418:126816, 2022) (see also Avram et al. in Appl Math Comput 423:127012, 2022), this method allows one to consider more general classes of costs; in particular state-dependent ones. On the other hand, the paper is completed by linear programming methods allowing one to deal with possibly discontinuous costs. In particular, we propose a brief exposition of classes of linearized dynamic programming principles based on our previous work and ensuing dual linear programming algorithms. We emphasize the particularities of our state space and possible generations of forward scenarios using the description of reachable sets.

Insights into the Core of the Assignment Game Via Complementarity

The classic paper of Shapley and Shubik \cite{Shapley1971assignment} characterized the core of the assignment game using ideas from matching theory and LP-duality theory and their highly non-trivial interplay. The worth of this game is given by an optimal solution to the primal LP and core imputations correspond to optimal solutions to the dual LP. This fact naturally raises the question of viewing core imputations through the lens of complementarity. Our exploration along these lines yields new insights: we obtain a relationship between the competitiveness of individuals, and teams of agents, and the amount of profit they accrue. It also sheds light on the phenomenon of degeneracy, i.e., when the optimal assignment is not unique. Shapley and Shubik's suggestion for dealing with degeneracy was to perturb the edge weights of the underlying graph to make the optimal assignment unique; however, this destroys crucial information contained in the original instance and the outcome becomes a function of the vagaries of the randomness imposed on the instance. The core is a quintessential solution concept in cooperative game theory. It contains all ways of distributing the total worth of a game among agents in such a way that no sub-coalition has incentive to secede from the grand coalition.

Computational aspects of infeasibility analysis in mixed integer programming

The analysis of infeasible subproblems plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems: conflict graph analysis and dual proof analysis. While conflict graph analysis detects sets of contradicting variable bounds in an implication graph, dual proof analysis derives valid linear constraints from the proof of the dual LP’s unboundedness. The main contribution of this paper is twofold. Firstly, we present three enhancements of dual proof analysis: presolving via variable cancellation, strengthening by applying mixed integer rounding functions, and a filtering mechanism. Further, we provide a comprehensive computational study evaluating the impact of every presented component regarding dual proof analysis. Secondly, this paper presents the first combined approach that uses both conflict graph and dual proof analysis simultaneously within a single MIP solution process. All experiments are carried out on general MIP instances from the standard public test set Miplib 2017; the presented algorithms have been implemented within the non-commercial MIP solver SCIP and the commercial MIP solver FICO Xpress.

Dual Circularly Polarized Fabry–Perot Resonator Antenna Employing a Polarization Conversion Metasurface

A dual circularly polarized (CP) Fabry-Perot (FP) resonator antenna is presented in this paper. A linear to circular polarization conversion metasurface (MS) is employed as the superstrate of the FP resonator antenna. The working principle of the unit cell adopted by the MS enables it to independently control the transmission and reflection characteristics. Firstly, the MS is designed to present high reflectivity to ensure the high gain of the FP resonator antenna. Then the capability of polarization conversion is endowed to it. The square patch of the unit cell can receive two orthogonal LP waves, and the cross slot on the ground plane can also make them get through. Meanwhile, the circular patch with 45° tilted etched slot can radiate different CP waves when excited by different LP waves. Therefore, the proposed MS can combine the function of partial reflection and transmission polarization conversion. And for different LP waves, the MS can convert them into different CP waves. A dual LP microstrip patch antenna is designed to act as the feed of the resonator antenna. When the antenna is fed at different ports, different LP waves are excited in the resonant cavity and are converted into different CP waves by the MS, thus the antenna obtains dual CP radiation. The measured results indicate that the antenna can exhibit excellent dual CP radiation characteristic. Compared with other reported dual CP resonator antennas, the proposed antenna can obtain well performance with simple structure and low profile.

Broadband and Low-Profile Penta-Polarization Reconfigurable Metamaterial Antenna

In this paper, a penta-polarization reconfigurable antenna with mushroom-type metamaterial loading is proposed, which can operate in $x$ -direction linear polarization ( $x$ -LP), $y$ -LP, 45°-LP, left-hand circular polarization (LHCP) and right-hand circular polarization (RHCP). In order to realize polarization reconfigurability, a dual-port dual LP mushroom antenna excited by crossed H-shape slot is designed by characteristic mode analysis (CMA). Through shifting the states of PIN diodes on the reconfigurable feeding network, the amplitude and phase distributions of two ports of the dual LP antenna can be dynamically controlled and the five polarization modes can be achieved. A prototype with a profile of $0.06\lambda _{0}$ at 5.2 GHz ( $\lambda _{0}$ is the operated wavelength in free space) is simulated and measured. The measurement results exhibit a wide impedance bandwidth over 33.2% for LPs and over 38.7% for CPs. In addition, the measured axial ratio bandwidths are 28.0% for LHCP and 30.3% for RHCP, and the maximum measured boresight gains of all modes are higher than 8.2 dBi. The simulated and measured results verify that the proposed antenna can achieve a balance between low-profile and broadband on the basis of penta-polarization reconfigurability. The proposed antenna can be applied to polarization diversity and C band satellite communication.

On Consequences of the Strong Convergence in Lebesgue–Orlich Spaces

We study the continuity in the sense of the strong topology for the flux function υ → l(υ) = |υ|p(⋅) − 2υ acting from the Lebesgue–Orlicz space Lp(⋅)(Ω, ℝm) to the dual Lp ′ (⋅)(Ω, ℝm), where p′(⋅) is the Holder-conjugate exponent, under the assumption that p(·) is an L∞(Ω)-function such that 1 < α ≤ p(·) ≤ β < ∞. We obtain estimates for the convergence $$ {\left\Vert l\left({\upsilon}_n\right)-l\left(\upsilon \right)\right\Vert}_{p^{\prime}\left(\cdot \right)}\to 0 $$ with respect to the smallness order as ‖υn − υ‖p(⋅) → 0. The strong continuity of the energy functional $$ \underset{\varOmega }{\int }{\left|\upsilon \right|}^{p\left(\cdot \right)} dx $$ is a consequence of the strong continuity of the flux function.

Approximation Algorithms for Movement Repairmen

In the Movement Repairmen (MR) problem, we are given a metric space ( V , d ) along with a set R of k repairmen r 1 , r 2 , …, r k with their start depots s 1 , s 2 , …, s k ∈ V and speeds v 1 , v 2 , …, v k ⩾ 0, respectively, and a set C of m clients c 1 , c 2 , …, c m having start locations s ′ 1 , s ′ 2 , …, s ′ m ∈ V and speeds v ′ 1 , v ′ 2 , …, v ′ m ⩾ 0, respectively. If t is the earliest time a client c j is collocated with any repairman (say, r i ) at a node u , we say that the client is served by r i at u and that its latency is t . The objective in the (S um -MR) problem is to plan the movements for all repairmen and clients to minimize the sum (average) of the clients’ latencies. The motivation for this problem comes, for example, from Amazon Locker Delivery [Amazon 2010] and USPS gopost [Service 2010]. We give the first O (log n )-approximation algorithm for the S um -MR problem. In order to approximate S um -MR, we formulate an LP for the problem and bound its integrality gap. Our LP has exponentially many variables; therefore, we need a separation oracle for the dual LP. This separation oracle is an instance of the Neighborhood Prize Collecting Steiner Tree (NPCST) problem in which we want to find a tree with weight at most L collecting the maximum profit from the clients by visiting at least one node from their neighborhoods. The NPCST problem, even with the possibility to violate both the tree weight and neighborhood radii, is still very hard to approximate. We deal with this difficulty by using LP with geometrically increasing segments of the timeline, and by giving a tricriteria approximation for the problem. The rounding needs a relatively involved analysis. We give a constant approximation algorithm for S um -MR in Euclidean Space where the speed of the clients differs by a constant factor. We also give a constant approximation for the makespan variant.

Comparative analysis of the affine scaling and Karmarkar’s polynomial – time for linear programming

The simplex method is the well-known, non-polynomial solution technique for linear programming problems. However, some computational testing has shown that the Karmarkar’s polynomial projective interior point method may perform better than the simplex method on many classes of problems, especially, on problems with large sizes. The affine scaling algorithm is a variant of the Karmarkar’s algorithms. In this paper, we compare the affine scaling and the Karmarkar algorithms using the same test LP problem. Keywords: Polynomial-time, Complexity bound, Primal LP, Dual LP, Basic Solution, Degenerate Solution, Affine Space, Simplex and Polytope

Computing shadow prices/costs of degenerate LP problems with reduced simplex tables

In applications of linear programming, shadow prices/costs are as important as the optimal values of decision variables and objective function. When the linear programming problems are primal degenerate, the shadow prices are no longer necessarily equal to optimal value of dual variables. In such cases, the so-called two-sided shadow prices are defined. However, existing approaches for two-sided shadow prices are tedious and unnoticed by decision-makers. The situation for shadow costs in a dual degenerate LP problem is the same. This study will first review and generalize the approaches of shadow prices in related studies, and then propose an easy approach to compute the shadow prices with respect to a resource and/or a resource bundle by using a reduced optimal simplex table. With slight modification, the proposed approach can also find the shadow costs with respect to an activity and/or an activity bundle. Numerical examples are used to illustrate the new approaches. Furthermore, some other important topics are discussed, as: the paradoxical situation, complementary effect between resources, weakly redundant constraint and the optimal change vector. We believe the proposed approaches are efficient and useful not only in classroom teaching but also in software programming of commercial packages.

Uniform LP duality for semidefinite and semi-infinite programming

Recently, a semidefinite and semi-infinite linear programming problem (SDSIP), its dual (DSDSIP), and uniform LP duality between (SDSIP) and (DSDSIP) were proposed and studied by Li et al. (Optimization 52:507–528, 2003). In this paper, we show that (SDSIP) is an ordinary linear semi-infinite program and, therefore, all the existing results regarding duality and uniform LP duality for linear semi-infinite programs can be applied to (SDSIP). By this approach, the main results of Li et al. (Optimization 52:507–528, 2003) can be obtained easily.

Symmetric approximate linear programming for factored MDPs with application to constrained problems

A weakness of classical Markov decision processes (MDPs) is that they scale very poorly due to the flat state-space representation. Factored MDPs address this representational problem by exploiting problem structure to specify the transition and reward functions of an MDP in a compact manner. However, in general, solutions to factored MDPs do not retain the structure and compactness of the problem representation, forcing approximate solutions, with approximate linear programming (ALP) emerging as a promising MDP-approximation technique. To date, most ALP work has focused on the primal-LP formulation, while the dual LP, which forms the basis for solving constrained Markov problems, has received much less attention. We show that a straightforward linear approximation of the dual optimization variables is problematic, because some of the required computations cannot be carried out efficiently. Nonetheless, we develop a composite approach that symmetrically approximates the primal and dual optimization variables (effectively approximating both the objective function and the feasible region of the LP), leading to a formulation that is computationally feasible and suitable for solving constrained MDPs. We empirically show that this new ALP formulation also performs well on unconstrained problems.

Nonlinear Rescaling as Interior Quadratic Prox Method in Convex Optimization

A class Ψ of strictly concave and twice continuously differentiable functions ψ: R → R with particular properties is used for constraint transformation in the framework of a Nonlinear Rescaling (NR) method with “dynamic” scaling parameter updates. We show that the NR method is equivalent to the Interior Quadratic Prox method for the dual problem in a rescaled dual space.The equivalence is used to prove convergence and to estimate the rate of convergence of the NR method and its dual equivalent under very mild assumptions on the input data for a wide class Ψ of constraint transformations. It is also used to estimate the rate of convergence under strict complementarity and under the standard second order optimality condition.We proved that for any ψ ∈ Ψ, which corresponds to a well-defined dual kernel ϕ = −ψ*, the NR method applied to LP generates a quadratically convergent dual sequence if the dual LP has a unique solution.

Dual LP model solution of an inventory problem using dynamic programming technique

Dual LP model solution of an inventory problem using dynamic programming technique

Dual Lp affine isoperimetric inequalities

We establish some inequalities for the dual -centroid bodies which are the dual forms of the results by Lutwak, Yang, and Zhang. Further, we establish a Brunn-Minkowski-type inequality for the polar of dual -centroid bodies.

Optimum workforce scheduling under the (14, 21) days-off timetable

An efficient optimum solution is presented for a real-life employee day-soff scheduling problem with a three-week cycle. Over a given work cycle, each worker is given 14 successive workdays and 7 successive off days. This three-week days-off timetable is referred to as the (14, 21) schedule. Given different labor demands for each day of the week, the primary objective is to minimize the number of workers. The secondary objective is to reduce transportation cost by minimizing the number of active days-off patterns. The solution technique utilizes the dual LP solution to determine the minimum number of workers and feasible days-off assignments, without using linear or integer programming. The simple solution technique eliminates the need to use integer programming for this particular scheduling problem.

Experimental and Theoretical Studies of a Novel Venturi Lean Premixed Prevaporized (LPP) Combustor

A new gas turbine engine using a unique layout patented in Norway has a low-emission combustion system under development. The gas generator uses entirely radial rotating components and employs a dual entry LP radial compressor, a radial HP compressor, and a radial HP turbine. The power turbine is of a two-stage axial design, coupled to an epicyclical gear embedded in the exhaust duct. Several combustor concepts have been tested and evaluated during the development of the engine. The engine is targeted for marine, power generation, and train propulsion. For the marine and train application liquid fuel operation is needed, thus the primary focus in the development has been for a lean premixed prevapourised system. An interesting concept utilizing two venturi premixers has been studied intensively. By utilizing venturi premixers the following advantages can be achieved: (1) low overall pressure drop but high injector pressure drop and velocities in the mixing region (throat region), (2) high shear forces and drag imposed on the droplets enhancing droplet shedding and evaporation, and (3) excellent emission behavior at designated load conditions. Although these advantages can benefit gas turbine low-emission combustion, the challenges in using venturi premixers are: (1) venturis are susceptible to separation and thus flame stabilization within the venturi which is detrimental and (2) inlet flow disturbances enhance the tendency for separation in the venturis and must be minimized. Studies were launched to investigate a proposed combustor configuration. These studies included analytical studies, computational fluid dynamics (CFD) calculations of isothermal and combusting flow inside the combustor together with rig tests at atmospheric, medium, and full pressure. Finally, engine tests within the full operating range were conducted with very favorable emission figures for lean premixed prevaporized (LPP) operation. The system was capable of running at below 20 ppm NOx and CO, at elevated power for liquid fuel. Control of part load performance and emissions is by variable fuel staging of the two venturi stages. The paper highlights the features of the venturi combustor development and discusses the characteristics in terms of flow conditions and droplet motion, heat transfer, ignition delay time, and emissions.

Optimal static voltage stability improvement using a numerically stable SLP algorithm, for real time applications

A numerically stable sequential Primal–Dual LP algorithm for the reactive power optimisation (RPO) is presented in this article. The algorithm minimises the voltage stability index C 2 [1] of all the load buses to improve the system static voltage stability. Real time requirements such as numerical stability, identification of the most effective subset of controllers for curtailing the number of controllers and their movement can be handled effectively by the proposed algorithm. The algorithm has a natural characteristic of selecting the most effective subset of controllers (and hence curtailing insignificant controllers) for improving the objective. Comparison with transmission loss minimisation objective indicates that the most effective subset of controllers and their solution identified by the static voltage stability improvement objective is not the same as that of the transmission loss minimisation objective. The proposed algorithm is suitable for real time application for the improvement of the system static voltage stability.

Sensible Rules for Remembering Duals—the S-O-B Method

We present a natural motivation and simple mnemonic for creating the dual LP of any linear program- ming problem.