Linear Programming Algorithm Research Articles

Posterior Constraint Selection for Nonnegative Linear Programming

Posterior constraint optimal selection techniques (COSTs) are developed for nonnegative linear programming problems (NNLPs), and a geometric interpretation is provided. The posterior approach is used in both a dynamic and non-dynamic active-set framework. The computational performance of these methods is compared with the CPLEX standard linear programming algorithms, with two most-violated constraint approaches, and with previously developed COST algorithms for large-scale problems.

Research on scheduling optimisation for an integrated system of wind‐photovoltaic‐hydro‐pumped storage

Wind power and photovoltaic power is increasingly becoming the most promising new energy with its advantages of rich resources, environmental protection, and low-carbon characters. However, with the large scale of wind power and photovoltaic power generation integrated to the power grid, it's random, intermittent and fluctuant output brings great challenge to the safety and stability of power grid. To ensure the security and stability of the system, wind power and photovoltaic power should be abandoned sometimes; this limits the new energy acceptance ability of the power grid. However, wind power, photovoltaic power and hydropower have the natural complementary features, utilising the complementary characteristic of the above three energy can make the output power curve smoother. Thus, this study combines the wind power, photovoltaic power, hydropower and pumped storage power to form a complementary generation system. In addition, this study proposes two optimal scheduling models of multi-energy power generation. The first model aims to minimise the cost of thermal units, the second model aims to minimise the fluctuation of the complementary generation system. Last, this study uses a computationally available mixed-integer linear programming algorithm to solve the model. A test system is used to achieve the optimal economic dispatch and verify the rationality and feasibility of the proposed model. The results illustrate the efficiency of the proposed strategy in improving the utilisation of renewable energy as well as reducing the impact of fluctuation due to wind and photovoltaic integrated to the power grid.

Optimizing Public Transport Systems in Sub-saharan Africa Using Operational Research Technique: A Focus on Nigeria

The rapid urbanization of sub-Saharan Africa metropolitan areas combined with inadequate or poorly executed development plans, has given rise to numerous transportation problems, including deteriorated physical attractions and comfort of road based public transport. This addresses these problems. Mathematical model is used to optimize the public transport system of Lagos, Nigeria. Lagos being a typical sub-Saharan metropolitan city, is used as a good representation of others. The public transport system in Lagos is modelled using linear programming algorithm. The resulting linear programming model was solved using simplex method, with the aid of a computer software – LIP solver. The optimal transport system, if implemented, will eradicate the problem caused by poor public transport system in sub-Saharan Africa or at least reduce them to the barest minimum.

On the Reliability of Optimization Results for Trigeneration Systems in Buildings, in the Presence of Price Uncertainties and Erroneous Load Estimation

Cogeneration and trigeneration plants are widely recognized as promising technologies for increasing energy efficiency in buildings. However, their overall potential is scarcely exploited, due to the difficulties in achieving economic viability and the risk of investment related to uncertainties in future energy loads and prices. Several stochastic optimization models have been proposed in the literature to account for uncertainties, but these instruments share in a common reliance on user-defined probability functions for each stochastic parameter. Being such functions hard to predict, in this paper an analysis of the influence of erroneous estimation of the uncertain energy loads and prices on the optimal plant design and operation is proposed. With reference to a hotel building, a number of realistic scenarios is developed, exploring all the most frequent errors occurring in the estimation of energy loads and prices. Then, profit-oriented optimizations are performed for the examined scenarios, by means of a deterministic mixed integer linear programming algorithm. From a comparison between the achieved results, it emerges that: (i) the plant profitability is prevalently influenced by the average “spark-spread” (i.e., ratio between electricity and fuel price) and, secondarily, from the shape of the daily price profiles; (ii) the “optimal sizes” of the main components are scarcely influenced by the daily load profiles, while they are more strictly related with the average “power to heat” and “power to cooling” ratios of the building.

Geometric random edge

We show that a variant of the random-edge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs $$\max \{c^Tx :x \in \mathbb R^n, \, Ax\leqslant b\}$$max{cTx:xźRn,Axźb}, whose constraint matrix A satisfies a geometric property introduced by Brunsch and Roglin: The sine of the angle of a row of A to a hyperplane spanned by $$n-1$$n-1 other rows of A is at least $$\delta $$ź. This property is a geometric generalization of A being integral and each sub-determinant of A being bounded by $$\Delta $$Δ in absolute value. In this case $$\delta \geqslant 1/(\Delta ^2 n)$$źź1/(Δ2n). In particular, linear programs defined by totally unimodular matrices are captured in this framework. Here $$\delta \geqslant 1/ n$$źź1/n and Dyer and Frieze previously described a strongly polynomial-time randomized simplex algorithm for linear programs with A totally unimodular. The expected number of pivots of the simplex algorithm is polynomial in the dimension and $$1/\delta $$1/ź and independent of the number of constraints of the linear program. Our main result can be viewed as an algorithmic realization of the proof of small diameter for such polytopes by Bonifas et al., using the ideas of Dyer and Frieze.

Topology optimization of embedded piezoelectric actuators considering control spillover effects

This article addresses the problem of active structural vibration control by means of embedded piezoelectric actuators. The topology optimization method using the solid isotropic material with penalization (SIMP) approach is employed in this work to find the optimum design of actuators taken into account the control spillover effects. A coupled finite element model of the structure is derived assuming a two-phase material and this structural model is written into the state-space representation. The proposed optimization formulation aims to determine the distribution of piezoelectric material which maximizes the controllability for a given vibration mode. The undesirable effects of the feedback control on the residual modes are limited by including a spillover constraint term containing the residual controllability Gramian eigenvalues. The optimization of the shape and placement of the conventionally embedded piezoelectric actuators are performed using a Sequential Linear Programming (SLP) algorithm. Numerical examples are presented considering the control of the bending vibration modes for a cantilever and a fixed beam. A Linear-Quadratic Regulator (LQR) is synthesized for each case of controlled structure in order to compare the influence of the additional constraint.

The optimal feedrate planning on five-axis parametric tool path with geometric and kinematic constraints for CNC machine tools

The optimal feedrate planning on five-axis parametric tool path with multi-constraints remains challenging due to the variable curvature of tool path curves and the nonlinear relationships between the Cartesian space and joint space. The methods for solving this problem are very limited at present. The optimal feedrate associated with a programmed tool path is crucial for high speed and high accuracy machining. This paper presents a novel feedrate optimisation method for feedrate planning on five-axis parametric tool paths with preset multi-constraints including chord error constraint, tangential kinematic constraints and axis kinematic constraints. The proposed method first derives a linear objective function for feedrate optimisation by using a discrete format of primitive continuous objective function. Then, the preset multi-constraints are converted to nonlinear constraint conditions on the decision variables in the linear objective function and are then linearised with an approximation strategy. A linear model for feedrate optimisation with preset multiple constraints is then constructed, which can be solved by well-developed linear programming algorithms. Finally, the optimal feedrate can be obtained from the optimal solution and fitted to the smooth spline curve as the ultimate feedrate profile. Experiments are conducted on two parametric tool paths to verify the feasibility and applicability of the proposed method that show both the planning results and computing efficiency are satisfactory when the number of sampling positions is appropriately determined.

Investigation of Ready Mixed Concrete Transportation Problem Using Linear Programming and Genetic Algorithm

Ready-mixed concrete (RMC) is one of the most common building material for construction industry for nearly all developed and developing countries. Generally, because of the technical requirements, concrete must be mixed in a batch plant and transported to the construction site. There are two important factors affected the cost of RMC: raw material cost and transportation cost. Additionally, transportation cost is also included when determining the unit price of RMC. However, profitability affects adversely in the case of long distance between the plant and construction site. For these reason, distribution of RMC from supply to the demand points with minimum cost is aimed in this study. This work contributes to both modelling and dispatching of RMC as an optimization problem by applying linear and heuristic methods. For this purpose, as an example, an urban area which divided into 7 districts and contained 4 concrete batch plants is discussed. Linear programming and genetic algorithm were applied to solve this problem and compared each other under the same conditions. The result shows linear programming is more efficient for this application because of the limited constraints and variables.

Completely mixed strategies for single controller unichain semi-Markov games with undiscounted payoffs

Zero-sum two-person finite undiscounted (limiting ratio average) semi-Markov games are considered where the transition probabilities and the transition times are controlled by a fixed player in all states. We prove that if such a game is unichain, the value and optimal stationary strategies can be obtained from an optimal solution of a linear programming algorithm for the associated undiscounted unichain single controller stochastic game (obtained by a data transformation method). The single controller undiscounted unichain semi-Markov games have been formulated as a linear complementarity problem and solved using a stepwise principal pivoting algorithm. We provide necessary and sufficient conditions for such games to be completely mixed (i.e., every optimal stationary strategy for each player assigns a positive probability to every action in every state). Some properties analogous to completely mixed matrix games are also established in this paper.

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.

Hardware/Software Partitioning for Heterogenous MPSoC Considering Communication Overhead

Hardware/software partitioning (HSP) is an important step in the co-design of hardware/software. This paper addresses the problem of HSP with communication (HSPC) on heterogeneous multiprocessor system-on-chip (MPSoC). The classic HSP is modeled as an optimization problem with an objective of minimizing the finishing time in system under the hardware area constraints. First, we put forward an optimal method, integer linear programming (ILP) algorithm, for solving the problem for small inputs. Then, we propose two other algorithms using dynamic programming (DP) method, i.e., Optimal Tree Partitioning (OTP) method for tree-structured graphs and Tree Cover Partitioning (TCP) algorithm for general graphs in polynomial time. The overall performance of the proposed algorithms is evaluated through comparisons with that of a genetic algorithm (GA) and a greedy algorithm which are commonly used to solve HSP problem. We have conducted experimental performance evaluation on various benchmarks with different combinations of computation to communication ratios and hardware area constraints. The experimental results show that OTP algorithm can generate optimal solutions with much faster speed than ILP, and TCP algorithm can obtain near-optimum with higher quality than those produced by GA and greedy algorithm.

Adaptive feedrate planning on parametric tool path with geometric and kinematic constraints for CNC machining

Feedrate planning is crucial for CNC systems to generate a smooth and fast movement, which is closely related to the machining quality and machining efficiency. In this paper, a new method for adaptive feedrate planning is proposed for parametric tool path with respecting geometric and kinematic constraints for CNC machining. The mathematic relationships between the prescribed constraints and tool tip feedrate are first formulated. Then the feedrate upper bound at any curve parameter is derived, and the tool path is adaptively discretized into discrete sampling positions and corresponding maximum feedrates are calculated. A linear mathematic model for linear programming is proposed to obtain the optimal square values of feedrates at each sampling position. After solving the linear model by a well-developed linear programming algorithm, the optimal feedrates at each sampling position can be calculated. Finally, the optimal discrete feedrate data are fitted to a smooth spline curve as the ultimate feedrate profile, which respects all prescribed constraints. Experiments are conducted on two parametric tool paths to show the merits of the proposed method. Results show that the proposed method are feasible and applicable, the generated feedrate profiles of the two tool paths respect all the prescribed constraints.

Partial Train Speed Trajectory Optimization Using Mixed-Integer Linear Programming

The inexorable increase in energy demand around the world has put the energy-saving technology in hot spot for railway transportation. Train speed trajectory optimization based on optimal control, coasting control, and collaborative control inside railway systems is a popular methodology to enhance energy efficiency. This paper studies a special and interesting problem, i.e., the partial train speed trajectory optimization problem, and proposes a complete mathematical model where a mixed-integer linear programming algorithm can be directly applied. During the transient operation process of a train, the speed of the train is often considered to be monotonically increasing and decreasing in normal conditions without extreme gradients. Given that, the proposed method can quickly locate the train speed profile under practical engineering constraints, and the objective function is either to maximize the regenerative braking energy or to minimize the traction energy. Such a method with a short computational time may become particularly interesting for online cases where a train is altering its speed in a fixed distance and time due to the operational requirement. The generated speed trajectory can be used to guide the train to control its speed or in a normal braking operation. The robustness and effectiveness of the method has been demonstrated through a number of detailed simulation results in this paper.

Abstractive Cross-Language Summarization via Translation Model Enhanced Predicate Argument Structure Fusing

Cross-language multidocument summarization is the task to generate a summary in a target language (e.g., Chinese) from a collection of documents in a different source language (e.g., English). Previous methods such as the extractive and compressive algorithms focus only on single sentence selection and compression, which cannot make full use of the similar sentences containing complementary information. Furthermore, the translation model knowledge is not fully explored in previous approaches. To address these two problems, we propose in this paper an abstractive cross-language summarization framework. First, the source language documents are translated into target language with a machine translation system. Then, the method constructs a pool of bilingual concepts and facts represented by the bilingual elements of the source-side predicate-argument structures (PAS) and their target-side counterparts. Finally, new summary sentences are produced by fusing bilingual PAS elements with the integer linear programming algorithm to maximize both of the salience and translation quality of the PAS elements. The experimental results on English-to-Chinese cross-language summarization demonstrate that our proposed method outperforms the state-of-the-art extractive systems in both automatic and manual evaluations.

An Efficient and Effective Methodology to Control Turn-On Sequence of Power Switches for Power Gating Designs

As technology advances, power consumption becomes a big challenge in modern very large-scale integration designs. To resolve this problem, power-gated technology has been widely adopted in circuit designs. Since the turn-on sequence of power switches has a great impact on the rush current, wake-up, and sequence times of a power gating design, this paper proposes a methodology to construct a hybrid routing structure to connect power switches. Our hybrid routing structure can induce less rush current and satisfy timing constraints because a better daisy chain is constructed. To find members of the daisy chain, an integer linear programming algorithm is used to pick up suitable power switches. To determine suitable depth of a daisy chain, we propose a model for a power gating design and induct precise equations to estimate voltage and rush current equations according to the model. All of our experiments are based on industrial designs and measured by vendor tools. The experimental results demonstrate the efficiency and effectiveness of our design methodology.

Co-Optimization of Storage System Sizing and Control Strategy for Intelligent Photovoltaic Power Plants Market Integration

Energy storage systems (ESS) when integrated with large-scale photovoltaic (PV) plants, constituting a so-called Intelligent PV (IPV) power plant, are able to contribute to improve the economic viability of these power plants and to help them participate in the electricity markets as other traditional generators. In this paper, the sizing and control strategy co-optimization for an existing IPV power plant is proposed and implemented. A global linear programming (LP) optimization algorithm is developed, where the optimal components sizing is computed directly in the same optimization as the operating management of the storage system. In the IPV power plant design stage, the LP optimization is applied to obtain the optimal energy storage sizing parameters (capacity and power rate). In the operation stage, the same LP algorithm optimizes the control strategy to participate to electricity markets. This participation is based on the online model predictive control (MPC) to counteract PV forecast errors by participating in intraday markets. The proposed approach (optimal ESS sizing and online MPC) increases the daily economic benefits around 20%.

Optimistic optimization for model predictive control of max-plus linear systems

Model predictive control for max-plus linear discrete-event systems usually leads to a nonsmooth nonconvex optimization problem with real valued variables, which may be hard to solve efficiently. An alternative approach is to transform the given problem into a mixed integer linear programming problem. However, the computational complexity of current mixed integer linear programming algorithms increases in the worst case exponentially as a function of the prediction horizon. The focus of this paper is on making optimistic optimization suited to solve the given problem. Optimistic optimization is a class of algorithms that can find an approximation of the global optimum for general nonlinear optimization. A key advantage of optimistic optimization is that one can specify the computational budget in advance and guarantee bounds on the suboptimality with respect to the global optimum. We prove that optimistic optimization can be applied for the given problem by developing a dedicated semi-metric and by proving it satisfies the necessary requirements for optimistic optimization. Moreover, we show that the complexity of optimistic optimization is exponential in the control horizon instead of the prediction horizon. Hence, using optimistic optimization is more efficient when the control horizon is small and the prediction horizon is large.

Applying Hybrid Interval Linear Programming and Genetic Algorithm to Coordinate Distance and Directional Over-current Relays

Interval linear programming (ILP) is a powerful tool for modeling problems with uncertain and bounded parameters. In real power systems, planned/unplanned events lead to changes in network topology. The changes cause altered distribution and magnitude of fault currents. Since protective relays are usually coordinated based on the fault currents of the main topology, mal-operation of the relays in such situations is likely. By covering the uncertainties, this paper aims to achieve the robust coordination of distance and directional overcurrent relays (D&DOCRs), as the main protections of transmission/subtransmission systems. Generally, relay coordination is a complicated nonlinear optimization problem. The complexity increases drastically by increasing coordination constraints, due to considering uncertainties. In this study, the problem is modeled as an ILP problem. The salient advantage of using ILP is that the number of constraints remains the same as in the main network topology, which greatly improves solving performance. Finally, to overcome the nonlinearity of the problem, a hybrid genetic algorithm and ILP (HGA/ILP), as a new optimization algorithm, is proposed. The proposed approach is applied to the IEEE 14-bus test system, and the results show that ILP is a useful tool to obtain robust settings of D&DOCRs in a power system.

A superlinearly convergent wide-neighborhood predictor–corrector interior-point algorithm for linear programming

In this paper we propose a new predictor-corrector algorithm with superlinear convergence in a wide neighborhood for linear programming problems. We let the centering parameter in a predictor step is chosen adaptively, which is different from other algorithms in the same wide neighborhood. The choice is a key for the local convergence of the new algorithm. In addition, we use the classical affine scaling direction as a part in a corrector step, not in a predictor step, which contributes to the complexity result. We prove that the new algorithm has a polynomial complexity of \(O(\sqrt{n}L)\), and the duality gap sequence is superlinearly convergent to zero, under the assumption that the iterate points sequence is convergent. Finally, numerical tests indicate its effectiveness.

Tool path generation via the multi-criteria optimisation for flat-end milling of sculptured surfaces

A method of generating optimal tool paths for sculptured surface machining with flat-end cutters is presented in this paper. The inclination and tilt angles, as well as the feed directions of the cutter at each cutter contact point on a machining path are optimised as a whole so that the machining width of the tool path can be as large as possible, and concerns such as smooth cutter motion, gouging avoidance, scallop height and machining widths overlap are also considered when calculating a path. A multi-criteria tool path optimisation model is introduced, and it is converted into a single objective optimisation with the weighted sum method. The Differential Evolution (DE) algorithm is suitable for solving this highly non-linear problem. However, the searching process of the DE algorithm may be trapped in local minima due to large number of design variables. Therefore, an algorithm combining the DE algorithm and the sequence linear programming algorithm is developed to solve this optimisation model. The proposed method is applied to two freeform surfaces to illustrate its effectiveness.