A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

The main goals of this paper are to: i) relate two iteration-complexity bounds derived for the Mizuno-Todd-Ye predictor-corrector (MTY P-C) algorithm for linear programming (LP), and; ii) study the geometrical structure of the LP central path. The first iteration-complexity bound for the MTY P-C algorithm considered in this paper is expressed in terms of the integral of a certain curvature function over the traversed portion of the central path. The second iteration-complexity bound, derived recently by the authors using the notion of crossover events introduced by Vavasis and Ye, is expressed in terms of a scale-invariant condition number associated with m × n constraint matrix of the LP. In this paper, we establish a relationship between these bounds by showing that the first one can be majorized by the second one. We also establish a geometric result about the central path which gives a rigorous justification based on the curvature of the central path of a claim made by Vavasis and Ye, in view of the behavior of their layered least squares path following LP method, that the central path consists of $${\mathcal{O}}(n^2)$$ long but straight continuous parts while the remaining curved part is relatively “short”.

We study the properties of the weighted central paths in linear programming. We consider each path as the function of the parameter 0 where the value at D 0 corresponds to the limit point at the boundary of the feasible set. We calculate the recursive formulas for the central path derivatives of all orders valid at each 0. We establish the geometric growth of the derivatives and, consequently, the analyticity of the weighted central path at 0 . This paper also provides the analysis of limiting behavior of those projection operators which often appear in the interior point methods. The central trajectory is one of the most important concepts in the interior point methods for linear programming. The central trajectory (or the central path) is a distinguished curve that is interior to the feasible set and tends to an optimal point. Most interior points algorithms follow, or are related to, the central trajectory. Although the central path concept is essential for the design and analysis of algorithms relatively few papers study properties of this curve itself. For a review on this topic we refer to (3). Recently Zhao and Zhu (16) have studied some analytical properties of the central trajectory in the interior of the feasible set. They have established an upper bound for the high order derivatives of the weighted central path and estimated the convergence radius of the corresponding Taylor series. They have also pointed out the relationship of these properties with the complexity of a path following algorithm where the number of iterations is estimated by a "curvature integral". (See also (15).) Further, Guler (3) has analysed some limiting properties of the central trajectory derivatives with respect to the parameter >0. He has established the existence of finite limits for derivatives of all orders as tends to infinity (that is, the central path stays in the interior) and to zero (the central path tends to the boundary). These results have applications to the construction of polynomial- time algorithms which are also locally fast. A surprising result of this study is that all derivatives of order > 1t end to 0 as!1, which helps to explain why interior point methods for linear programming require so few iterations (see (3)).

Linear programming algorithms have been widely used in Decision Support Systems. These systems have incorporated linear programming algorithms for the solution of the given problems. Yet, the special structure of each linear problem may take advantage of different linear programming algorithms or different techniques used in these algorithms. This paper proposes a web-based DSS that assists decision makers in the solution of linear programming problems with a variety of linear programming algorithms and techniques. Two linear programming algorithms have been included in the DSS: (i) revised simplex algorithm and (ii) exterior primal simplex algorithm. Furthermore, ten scaling techniques, five basis update methods and eight pivoting rules have been incorporated in the DSS. All linear programming algorithms and methods have been implemented using MATLAB and converted to Java classes using MATLAB Builder JA, while the web interface of the DSS has been designed using Java Server Pages.

*† This paper proposes a new algorithm for actuator allocation problems, e.g., reaction wheel torque distribution and jet selection logic. The algorithm is based on an interior-point linear programming (LP) algorithm. Compared to the simplex LP algorithm, the interiorpoint LP algorithm exhibits several advantages at the cost of reasonably increased computation. They encompasses greater robustness to uncertainty in actuator models, smoother actuation command over time, and greater flexibility in incorporating nonlinear operations such as quantization. These advantages stem from the fact that the interior algorithm searches for a suboptimal solution among the interior points of a feasible set and not among the extreme points or corner points of the feasible set as in the simplex algorithm. The proposed approach has been applied to the problem of reaction wheel torque distribution for the attitude control of an autonomous rendezvous and docking (ARD) robotic spacecraft and demonstrated to yield more effective solutions for the actuator allocation problem than a simplex LP approach.

The main viewpoint of land use planning is to resolve the conflict between competing land uses in one place that is carried out based on their economic, social and environmental suitability. Many different approaches have been designed to resolve this conflict but any of them have done. This study aimed to assess accuracy and validity compared four optimization algorithms, including linear programming (LP), simulated annealing (SA), multi-objective land use allocation (MOLA) and multidimensional choice (MDCHOICE) for land use planning in Gorgan Township. At first, land use allocation was done through the LP algorithm. Then, using achieved pixels quantity from LP optimization, the land uses was optimized in term of spatial suitability by other algorithms. The allocation maps generated through these algorithms were compared using statistical methods and landscape metrics. The map provided through MDCHOICE was used as a reference map and that generated through MOLA, SA and LP as a comparative one in a validation process. The similarities and agreements of the two maps were evaluated with various algorithms in IDRISI and MCK software. The results demonstrate that: (1) It follows that the LP algorithm to determine the optimal number of pixels to lead MDCHOICE has helped in the spatial allocation properly. (2) MOLA had a better performance for development in terms of all landscape metrics and SA algorithm was better in forestry land use allocation. Therefore the quality of spatial allocation both of them needs to be improved. (3) Although the LP algorithm is not superior to the algorithms based on the dimensions studied in this research, in terms of achieving the defined goals and the degree of deviation from the number of cells defined for each land use has the least violation. (4) The superiority of MDCHOICE over MOLA, SA and LP and applicability of various kappa statistics for comparing the maps for a more in depth analysis of agreements and disagreements.

The next generation (NextGen) transport aircraft configurations being investigated as part of the NASA Aeronautics Subsonic Fixed Wing Project have more control surfaces, or control effectors, than existing transport aircraft configurations. Conventional flight control is achieved through two symmetric elevators, two antisymmetric ailerons, and a rudder. The five control surfaces, reduced to three command variables, produce moments along the three main axes of the aircraft and enable the pilot to control the attitude of the aircraft. Next generation aircraft will have additional redundant control effectors to control the three moments, creating a situation where the aircraft is over-actuated and where a simple relationship no longer exists between the required control surface deflections and the desired moments. NextGen flight controllers will incorporate control allocation algorithms to determine the optimal effector commands to attain the desired moments, taking into account the effector limits. Approaches to solving the problem using linear programming and quadratic programming algorithms have been proposed and tested. It is of great interest to understand their relative advantages and disadvantages and how design parameters may affect their properties. In this paper, we investigate the sensitivity of the effector commands with respect to the desired moments and show on some examples the sensitivity of the solutions provided by the linear programming and quadratic programming methods.

Solving (mixed) integer (linear) programs, (M)I(L)Ps for short, is a fundamental optimisation task with a wide range of applications in artificial intelligence and computer science in general. While hard in general, recent years have brought about vast progress for solving structurally restricted, (non-mixed) ILPs: n-fold, tree-fold, 2-stage stochastic and multi-stage stochastic programs admit efficient algorithms, and all of these special cases are subsumed by the class of ILPs of small treedepth. In this paper, we extend this line of work to the mixed case, by showing an algorithm solving MILP in time f(a,d)poly(n), where a is the largest coefficient of the constraint matrix, d is its treedepth, and n is the number of variables. This is enabled by proving bounds on the denominators (fractionality) of the vertices of bounded-treedepth (non-integer) linear programs. We do so by carefully analysing the inverses of invertible sub-matrices of the constraint matrix. This allows us to afford scaling up the mixed program to the integer grid, and applying the known methods for integer programs. We then trace the limiting boundary of our "bounded fractionality" approach both in terms of going beyond MILP (by allowing non-linear objectives) as well as its usefulness for generalising other important known tractable classes of ILP. On the positive side, we show that our result can be generalised from MILP to MIP with piece-wise linear separable convex objectives with integer breakpoints. On the negative side, we show that going even slightly beyond such objectives or considering other natural related tractable classes of ILP leads to unbounded fractionality. Finally, we show that restricting the structure of only the integral variables in the constraint matrix does not yield tractable special cases.

We present observations on the linear programming algorithm proposed independently by Oliker and by Wang to design a single reflector for a point source and a far-field target. Given a set of source rays and discretized target intensities, the linear programming algorithm solves a variational problem to produce a reflector solution that consists of paraboloid patches. The computational complexity increases quadratically with the number of source rays and targets; this makes it important to minimize the number of source rays. However, minimizing the number of source rays results in solutions where the source rays at the intersection between neighboring patches split between multiple targets. This is unlike other discretized target reflector design methods, such as the Oliker supporting ellipsoid algorithm, that are used to aim numerous rays per ellipsoid at only one target. We uncovered a relationship between the optimal numbers of source rays and targets needed to run the problem. This relationship makes it possible to limit the number of source rays used in the design to the order of the number of target points. In this paper, we highlight the main features and current limitations of the linear programming algorithm. Finally, we propose a fast algorithm for 2D reflector design inspired by the intersection property of the linear programming method. The direct calculation method is shown to be several orders of magnitude faster than the linear programming method.

This paper studies the problem of tangible assets acquisition within the company by proposing a new hybrid model that uses linear programming and fuzzy numbers. Regarding linear programming, two methods were implemented in the model, namely: the graphical method and the primal simplex algorithm. This hybrid model is proposed for solving investment decision problems, based on decision variables, objective function coefficients, and a matrix of constraints, all of them presented in the form of triangular fuzzy numbers. Solving the primal simplex algorithm using fuzzy numbers and coefficients, allowed the results of the linear programming problem to also be in the form of fuzzy variables. The fuzzy variables compared to the crisp variables allow the determination of optimal intervals for which the objective function has values depending on the fuzzy variables. The major advantage of this model is that the results are presented as value ranges that intervene in the decision-making process. Thus, the company’s decision makers can select any of the result values as they satisfy two basic requirements namely: minimizing/maximizing the objective function and satisfying the basic requirements regarding the constraints resulting from the company’s activity. The paper is accompanied by a practical example.

It is well known that an arbitrary graphical model of statistical inference defined on a tree,i.e. on a graph without loops, is solved exactly and efficiently by an iterative beliefpropagation (BP) algorithm convergent to the unique minimum of the so-called Bethe freeenergy functional. For a general graphical model on a loopy graph, the functional may showmultiple minima, the iterative BP algorithm may converge to one of the minimaor may not converge at all, and the global minimum of the Bethe free energyfunctional is not guaranteed to correspond to the optimal maximum likelihood (ML)solution in the zero-temperature limit. However, there are exceptions to this generalrule, discussed by Kolmogorov and Wainwright (2005) and by Bayati et al (2006,2008) in two different contexts, where the zero-temperature version of the BPalgorithm finds the ML solution for special models on graphs with loops. Thesetwo models share a key feature: their ML solutions can be found by an efficientlinear programming (LP) algorithm with a totally uni-modular (TUM) matrix ofconstraints. Generalizing the two models, we consider a class of graphical modelsreducible in the zero-temperature limit to LP with TUM constraints. Assuming thata gedanken algorithm, g-BP, for finding the global minimum of the Bethe freeenergy is available, we show that in the limit of zero temperature, g-BP outputsthe ML solution. Our consideration is based on equivalence established betweengapless linear programming (LP) relaxation of the graphical model in the T → 0 limit and the respective LP version of the Bethe free energy minimization.

Connectionist networks for pivot selection in linear programming

A popular approach in combinatorial optimization is to model problems as integer linear programs. Ideally, the relaxed linear program would have only integer solutions, which happens for instance when the constraint matrix is totally unimodular. Still, sometimes it is possible to build an integer solution with same cost from the fractional solution. Examples are two scheduling problems [4,5] and the single disk prefetching/caching problem [3]. We show that problems such as the three previously mentioned can be separated into two subproblems: (1) finding an optimal feasible set of slots, and (2) assigning the jobs or pages to the slots. It is straigthforward to show that the latter can be solved greedily. We are able to solve the former with a totally unimodular linear program, from which we obtain simple combinatorial algorithms with improved worst case running time.

This paper gives an algorithm for solving linear programming problems. For a problem with n constraints and d variables, the algorithm requires an expected O(d 2 n) + (log n)O(d) d/2+O(1) + O(d 4 √nlog n) arithmetic operations, as n→∞ . The constant factors do not depend on d. Also, an algorithm is given for integer linear programming. Let φ bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector. Let n and d be as before. Then, the algorithm requires expected O(2 d dn + 8 d d√n ln ln n) + d O(d) φ ln n operations on numbers with d O(1) φ bits, as n→∞ , where the constant factors do not depend on d or φ to other convex programming problems. For example, an algorithm for finding the smallest sphere enclosing a set of n points in E d has the same time bound.

This paper proposes a novel energy management approach (imitation-Q-learning) based on imitation learning (IL) and reinforcement learning (RL). The proposed approach reinforces a decision-making agent based on a modified Q-learning algorithm to mimic an expert demonstration to solve a microgrid (MG) energy management problem. Those demonstrations are derived from solving a set of linear programming (LP) problems. Consequently, the imitation-Q-learning algorithm learns by interacting with the MG simulator and imitating the LP demonstrations to make decisions in real time that minimize the MG energy costs without prior knowledge of uncertainties related to photovoltaic (PV) production, load consumption, and electricity prices. A real-scale MG at the Lille Catholic University in France was used as a case study to conduct experiments. The proposed approach was compared to the expert performances, which are the LP algorithm and the conventional Q-learning algorithm in different test scenarios. It was approximately 80 times faster than conventional Q-learning and achieved the same performance as LP. In order to test the robustness of the proposed approach, a PV inverter crush and load shedding were also simulated. Preliminary results show the effectiveness of the proposed method.

The purpose of onboard thruster control allocation is to select specific thrusters and calculate their firing durations in order to realize force and torque commands derived from the control system of a spacecraft. This study is conducted to design an optimal thruster combinations table for the use of onboard real time control allocation problems. The algebraic method for generating the optimal thruster combinations table (OTCT) with respect to thruster configuration is explained in detail based on linear programming (LP). With the table pre-stored in the onboard computer, a simple but optimal algorithm to solve the thruster control allocation problem in real time is implemented. The proposed new algorithm, which has the same optimal solutions as LP algorithm, is much faster than LP in real time control allocation. Its effectiveness is demonstrated in the simulation.