Integer Quadratic Programming Problem Research Articles

MR&MR-SUM: MAXIMUM RELEVANCE AND MINIMUM REDUNDANCY DOCUMENT SUMMARIZATION MODEL

We have presented an approach to automatic document summarization. In the proposed approach, text summarization is modeled as a quadratic integer-programming problem. This model generally attempts to optimize three properties, namely, (1) relevance: summary should contain informative textual units that are relevant to the user; (2) redundancy: summaries should not contain multiple textual units that convey the same information; and (3) length: summary is bounded in length. To solve the optimization problem we have created a novel differential evolution algorithm. Experimental results on DUC2005 and DUC2007 data sets showed that the proposed approach outperforms the other methods.

Solving the canonical dual of box- and integer-constrained nonconvex quadratic programs via a deterministic direct search algorithm

This paper presents a massively parallel global deterministic direct search method (VTDIRECT) for solving nonconvex quadratic minimization problems with either box or±1 integer constraints. Using the canonical dual transformation, these well-known NP-hard problems can be reformulated as perfect dual stationary problems (with zero duality gap). Under certain conditions, these dual problems are equivalent to smooth concave maximization over a convex feasible space. Based on a perturbation method proposed by Gao, the integer programming problem is shown to be equivalent to a continuous unconstrained Lipschitzian global optimization problem. The parallel algorithm VTDIRECT is then applied to solve these dual problems to obtain global minimizers. Parallel performance results for several nonconvex quadratic integer programming problems are reported.

A Mixed Integer Quadratic Programming Model for the Low Autocorrelation Binary Sequence Problem

In this paper the low autocorrelation binary sequence problem (LABSP) is modeled as a mixed integer quadratic programming (MIQP) problem and proof of the model’s validity is given. Since the MIQP model is semidefinite, general optimization solvers can be used, and converge in a finite number of iterations. The experimental results show that IQP solvers, based on this MIQP formulation, are capable of optimally solving general/skew-symmetric LABSP instances of up to 30/51 elements in a moderate time. ACM Computing Classification System (1998): G.1.6, I.2.8.

Complete Solutions to Mixed Integer Programming

This paper considers a new canonical duality theory for solving mixed integer quadratic programming problem. It shows that this well-known NP-hard problem can be converted into concave maximization dual problems without duality gap. And the dual problems can be solved, under certain conditions, by polynomial algorithms.

Optimal Photovoltaic Array Reconfiguration to Reduce Partial Shading Losses

Partial shading of a photovoltaic array is the condition under which different modules in the array experience different irradiance levels due to shading. This difference causes mismatch between the modules, leading to undesirable effects such as reduction in generated power and hot spots. The severity of these effects can be considerably reduced by photovoltaic array reconfiguration. This paper proposes a novel mathematical formulation for the optimal reconfiguration of photovoltaic arrays to minimize partial shading losses. The paper formulates the reconfiguration problem as a mixed integer quadratic programming problem and finds the optimal solution using a branch and bound algorithm. The proposed formulation can be used for an equal or nonequal number of modules per row. Moreover, it can be used for fully reconfigurable or partially reconfigurable arrays. The improvement resulting from the reconfiguration with respect to the existing photovoltaic interconnections is demonstrated by extensive simulation results.

Adaptive bi-level programming for optimal gene knockouts for targeted overproduction under phenotypic constraints

BackgroundOptimization procedures to identify gene knockouts for targeted biochemical overproduction have been widely in use in modern metabolic engineering. Flux balance analysis (FBA) framework has provided conceptual simplifications for genome-scale dynamic analysis at steady states. Based on FBA, many current optimization methods for targeted bio-productions have been developed under the maximum cell growth assumption. The optimization problem to derive gene knockout strategies recently has been formulated as a bi-level programming problem in OptKnock for maximum targeted bio-productions with maximum growth rates. However, it has been shown that knockout mutants in fact reach the steady states with the minimization of metabolic adjustment (MOMA) from the corresponding wild-type strains instead of having maximal growth rates after genetic or metabolic intervention. In this work, we propose a new bi-level computational framework--MOMAKnock--which can derive robust knockout strategies under the MOMA flux distribution approximation.MethodsIn this new bi-level optimization framework, we aim to maximize the production of targeted chemicals by identifying candidate knockout genes or reactions under phenotypic constraints approximated by the MOMA assumption. Hence, the targeted chemical production is the primary objective of MOMAKnock while the MOMA assumption is formulated as the inner problem of constraining the knockout metabolic flux to be as close as possible to the steady-state phenotypes of wide-type strains. As this new inner problem becomes a quadratic programming problem, a novel adaptive piecewise linearization algorithm is developed in this paper to obtain the exact optimal solution to this new bi-level integer quadratic programming problem for MOMAKnock.ResultsOur new MOMAKnock model and the adaptive piecewise linearization solution algorithm are tested with a small E. coli core metabolic network and a large-scale iAF1260 E. coli metabolic network. The derived knockout strategies are compared with those from OptKnock. Our preliminary experimental results show that MOMAKnock can provide improved targeted productions with more robust knockout strategies.

Probabilistic-constrained optimal control of a class of stochastic hybrid systems

Stochastic hybrid systems have several applications such as biological systems and communication networks, but it is difficult to consider control of general stochastic hybrid systems. In this paper, a class of discrete-time stochastic hybrid systems, in which only discrete dynamics are stochastic, is considered. For this system, a solution method for the optimal control problem with probabilistic constraints is proposed. Probabilistic constraints guarantee that the probability that the continuous state reaches a given unsafe region is less than a given constant. In the propose method, first, continuous state regions, from which the state reaches a given unsafe region, are computed by a backward-reachability graph. Next, mixed integer quadratic programming problems with constraints derived from the backward-reachability graph are solved. The proposed method can be applied to model predictive control.

CDDS: Constraint-driven document summarization models

This paper proposes a constraint-driven document summarization approach emphasizing the following two requirements: (1) diversity in summarization, which seeks to reduce redundancy among sentences in the summary and (2) sufficient coverage, which focuses on avoiding the loss of the document’s main information when generating the summary. The constraint-driven document summarization models with tuning the constraint parameters can drive content coverage and diversity in a summary. The models are formulated as a quadratic integer programming (QIP) problem. To solve the QIP problem we used a discrete PSO algorithm. The models are implemented on multi-document summarization task. The comparative results showed that the proposed models outperform other methods on DUC2005 and DUC2007 datasets.

Biofuel supply chain design under competitive agricultural land use and feedstock market equilibrium

The rapid expansion of the biofuel industry diverts a large amount of agricultural crops as energy feedstocks, and in turn affects farm land allocation, feedstock market equilibrium, and agricultural economic development in local areas. In this paper, we propose game-theoretic models that incorporate farmers' decisions on land use and market choice into the biofuel manufacturers' supply chain design problem. A noncooperative bi-level Stackelberg leader–follower game model and a cooperative game model are developed respectively to address possible business partnership scenarios between feedstock suppliers and biofuel manufacturers. The models determine the optimal number and locations of biorefineries, the required prices for these refineries to compete for feedstock resources, as well as farmers' land use choices between food and energy. Using corn as an example of feedstock crops, spatial market equilibrium is utilized to model the relationship between corn supply and demand, and the associated price variations in local grain markets. With linear corn demand functions, we develop a solution approach that transforms the original discrete mathematical program with equilibrium constraints (DC-MPEC) into to a solvable mixed integer quadratic programming (MIQP) problem based on Karush–Kuhn–Tucker (KKT) conditions. The proposed methodology is illustrated using an empirical case study of the Illinois State. The computation results reveal interesting insights into optimal land use strategies and supply chain design for the emerging “biofuel economy”.

Solving Bilevel Programming Problems Using a Neural Network Approach and Its Application to Power System Environment

In this paper, a hybrid neural network approach to solve mixed integer quadratic bilevel programming problems is proposed. Bilevel programming problems arise when one optimization problem, the uppe...

Metabolic network analysis of perfused livers under fed and fasted states: Incorporating thermodynamic and futile-cycle-associated regulatory constraints

Isolated liver perfusion systems have been extensively used to characterize intrinsic metabolic changes in liver under various conditions, including systemic injury, hepatotoxin exposure, and warm ischemia. Most of these studies were performed utilizing fasted animals prior to perfusion so that a simplified metabolic network could be used in order to determine intracellular fluxes. However, fasting induced metabolic alterations might interfere with disease related changes. Therefore, there is a need to develop a “unified” metabolic flux analysis approach that could be similarly applied to both fed and fasted states. In this study we explored a methodology based on elementary mode analysis in order to determine intracellular fluxes and active pathways simultaneously. In order to decrease the solution space, thermodynamic constraints, and enzymatic regulatory properties for the formation of futile cycles were further considered in the model, resulting in a mixed integer quadratic programming problem. Given the published experimental observations describing the perfused livers under fed and fasted states, the proposed approach successfully determined that gluconeogenesis, glycogenolysis and fatty acid oxidation were active in both states. However, fasting increased the fluxes in gluconeogenic reactions whereas it decreased fluxes associated with glycogenolysis, TCA cycle, fatty acid oxidation and electron transport reactions. This analysis further identified that more pathways were found to be active in fed state while their weight values were relatively lower compared to fasted state. Glucose, lactate, glutamine, glutamate and ketone bodies were also found to be important external metabolites whose extracellular fluxes should be used in the hepatic metabolic network analysis. In conclusion, the mathematical formulation explored in this study is an attractive tool to analyze the metabolic network of perfused livers under various disease conditions. This approach could be simultaneously applied to both fasted and fed data sets.

Constructing uniform designs: A heuristic integer programming method

In this paper, the wrap-around L2-discrepancy (WD) of asymmetrical design is represented as a quadratic form, thus the problem of constructing a uniform design becomes a quadratic integer programming problem. By the theory of optimization, some theoretic properties are obtained. Algorithms for constructing uniform designs are then studied. When the number of runs n is smaller than the number of all level-combinations m, the construction problem can be transferred to a zero–one quadratic integer programming problem, and an efficient algorithm based on the simulated annealing is proposed. When n≥m, another algorithm is proposed. Empirical study shows that when n is large, the proposed algorithms can generate designs with lower WD compared to many existing methods. Moreover, these algorithms are suitable for constructing both symmetrical and asymmetrical designs.

Solutions to Periodic Sensor Scheduling Problems for Formation Flying Missions in Deep Space

A relaxation of the general infinite horizon sensor scheduling problem for state estimation in linear time-invariant systems is presented. The optimal schedule is assumed to be periodic, and the relaxation allows the full problem, a nonlinear combinatorial optimization problem, to be converted to a mixed integer quadratic programming problem which can be easily solved. The relaxation is based on the system being driven by low-intensity process noise. To find the solution the integer constraint is first relaxed to find an initial unconstrained solution using a standard quadratic programming solver. The integer constraint is then satisfied by using established integer- constrained quadratic least-squares techniques and the Hessian of the cost function. Examples illustrate the performance of the algorithm versus the truth solution found by an exhaustive search.

Global optimality conditions and optimization methods for quadratic integer programming problems

In this paper, we first establish some sufficient and some necessary global optimality conditions for quadratic integer programming problems. Then we present a new local optimization method for quadratic integer programming problems according to its necessary global optimality conditions. A new global optimization method is proposed by combining its sufficient global optimality conditions, local optimization method and an auxiliary function. The numerical examples are also presented to show that the proposed optimization methods for quadratic integer programming problems are very efficient and stable.

Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems

In this paper, we study a mixed integer constrained quadratic programming problem. This problem is NP-Hard. By reformulating the problem to a box constrained quadratic programming and solving the reformulated problem, we can obtain a global optimal solution of a sub-class of the original problem. The reformulated problem may not be convex and may not be solvable in polynomial time. Then we propose a solvability condition for the reformulated problem, and discuss methods to construct a solvable reformulation for the original problem. The reformulation methods identify a solvable subclass of the mixed integer constrained quadratic programming problem.

A note on “A new local and global optimization method for mixed integer quadratic programming problems” by G.Q. Li et al.

Abstract In this comment, we preset a minor mistake in typing which is made in “A new local and global optimization method for mixed integer quadratic programming problems” by G.Q. Li et al.

Nonintrusive appliance load monitoring based on integer programming

AbstractThis paper presents a new nonintrusive appliance load monitoring technique based on integer programming. Nonintrusive appliance load monitoring is the problem of identifying the operating conditions of electric appliances in a house by observing only the overall load current and voltage. Since the overall load current is expressed as a superposition of the currents of the operating appliances, the monitoring problem can be formulated as an integer quadratic programming problem by expressing the operating conditions as integer variables. This problem is solvable with a sufficiently small computational burden thanks to the recent development of commercial software. The proposed method does not require relearning even when a new appliance is installed in the house. Furthermore, the proposed formulation is applicable to cases in which some appliance has multiple modes, and cases in which some appliances of the same type are operating simultaneously. The usefulness of the proposed technique is verified by experimental results. © 2010 Wiley Periodicals, Inc. Electr Eng Jpn, 174(2): 18–25, 2011; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/eej.21040

Alternating direction augmented Lagrangian methods for semidefinite programming

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.

A new local and global optimization method for mixed integer quadratic programming problems

In this paper, a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable.

Convex relaxation and Lagrangian decomposition for indefinite integer quadratic programming1

We study lower bounding methods for indefinite integer quadratic programming problems. We first construct convex relaxations by D.C. (difference of convex functions) decomposition and linear underestimation. Lagrangian bounds are then derived by applying dual decomposition schemes to separable relaxations. Relationships between the convex relaxation and Lagrangian dual are established. Finally, we prove that the lower bound provided by the convex relaxation coincides with the Lagrangian bound of the orthogonally transformed problem. 1This article is dedicated to the memory of Professor Alex Rubinov.