Computational Experiments with Cross and Crooked Cross Cuts

In this paper, we study whether cuts obtained from two simplex tableau rows at a time can strengthen the bounds obtained by Gomory mixed-integer (GMI) cuts based on single tableau rows. We also study whether cross and crooked cross cuts, which generalize split cuts, can be separated in an effective manner for practical mixed-integer programs (MIPs) and can yield a nontrivial improvement over the bounds obtained by split cuts. We give positive answers to both these questions for MIPLIB 3.0 problems. Cross cuts are a special case of the t-branch split cuts studied by Li and Richard [Li Y, Richard J-PP (2008) Cook, Kannan and Schrijvers's example revisited. Discrete Optim. 5:724–734]. Split cuts are 1-branch split cuts, and cross cuts are 2-branch split cuts. Crooked cross cuts were introduced by Dash, Günlük, and Lodi [Dash S, Günlük O, Lodi A (2010) MIR closures of polyhedral sets. Math Programming 121:33–60] and were shown to dominate cross cuts by Dash, Günlük, and Molinaro [Dash S, Günlük O, Molinaro M (2012b) On the relative strength of different generalizations of split cuts. IBM Technical Report RC25326, IBM, Yorktown Heights, NY].United States. Office of Naval Research (Grant N000141110724

When Lift-and-Project Cuts are Different

In this paper, we present a method to determine if a lift-and-project cut for a mixed-integer linear program is irregular, in which case the cut is not equivalent to any intersection cut from the bases of the linear relaxation. This is an important question due to the intense research activity for the past decade on cuts from multiple rows of simplex tableau as well as on lift-and-project cuts from non-split disjunctions. While it is known since Balas and Perregaard (2003) that lift-and-project cuts from split disjunctions are always equivalent to intersection cuts and consequently to such multi-row cuts, Balas and Kis (2016) have recently shown that there is a necessary and sufficient condition in the case of arbitrary disjunctions: a lift-and-project cut is regular if, and only if, it corresponds to a regular basic solution of the Cut Generating Linear Program (CGLP). This paper has four contributions. First, we state a result that simplifies the verification of regularity for basic CGLP solutions from Balas and Kis (2016). Second, we provide a mixed-integer formulation that checks whether there is a regular CGLP solution for a given cut that is regular in a broader sense, which also encompasses irregular cuts that are implied by the regular cut closure. Third, we describe a numerical procedure based on such formulation that identifies irregular lift-and-project cuts. Finally, we use this method to evaluate how often lift-and-project cuts from simple $t$-branch split disjunctions are irregular, and thus not equivalent to multi-row cuts, on 74 instances of the MIPLIB benchmarks.Comment: INFORMS Journal on Computing (to appear

MODELING AND OPTIMIZATION OF THE MICROSPHERE GENERATION PROCESS

Microspheres (< 1000 μm) have applications in various fields (e.g., drug delivery, cosmetics, food, etc.). Microspheres can be generated by the micro-fluidic technique, in which microspheres are produced from one fluid under the action of another immiscible fluid in a network of channels. There are four performance indexes associated with a microsphere generation process with a device, namely (1) the size of microspheres (as small as possible), (2) the uniformity of size distributions (as high as possible), and (3) the flexibility of devices (i.e., the size range of microspheres that can be generated with one device), and (4) the efficacy of the microspheres generation process (mass production or not). Two operating principles along with their corresponding devices, the modified T-junction device and membrane emulsification device, are studied in this dissertation, because of their unique features, with the former having an excellent task flexibility and the latter having an excellent efficacy. The study defined three objectives, namely (1) understanding the mechanism of the microsphere generation process with the modified T-junction by both numerical investigation and experimental investigation, (2) optimizing the microsphere generation process with any micro-fluidic device in general and the modified T-junction device in particular (optimization: the size and uniformity), and (3) designing and fabricating a new emulsification membrane by tackling the shortcoming (i.e., fragile with the membrane) with the existing emulsification membrane. For objective (1), a simulation model was built first, validated by the experiment, and then the simulation model was employed to study the regimes. For objective (2), a new optimization procedure was first proposed for general micro-fluidic systems and then applied to the modified T-junction system. For objective (3), a new membrane was designed and fabricated and tested. The following conclusions can be drawn from the study: (1) the modified T-junction device works based on a combined operating principle (flow focusing and conventional T-junction) and there are three regimes (instead of the four regimes in the conventional T-junction) in the flow; (2) the optimization of the microsphere generation process makes sense for the micro-fluidic device in general and the modified T-junction in particular (the optimal modified T-junction is: the mean size: 16.1 μm and 24.8 μm, and the uniformity (Standard Deviation (SD)): 0.2 μm and 0.7 μm); (3) the shortcoming with emulsification membrane can be overcome with a multi-layer membrane architecture. There are several contributions made by this dissertation in the field of micro-fluidic. First is the provision of an accurate Computational Fluid Dynamics (CFD) model for the modified T-junction. Second is the new knowledge discovered regarding the mechanism of microsphere generation with the modified T-junction device. Third is the provision of an effective optimization approach for any micro-fluidic device in general and for the modified T-junction device in particular. Fourth is the design with the successful fabrication of the membrane emulsification device based on new system architecture (i.e., multi-layer structure). From an application’s perspective, this dissertation has provided evidence that with the micro-fluidic technique, the smallest size of microspheres can be 2.3 μm; the highest uniformity (SD) can be 0.8 μm. Further, if an application puts emphasis on the task flexibility, the modified T-junction device is an excellent choice, and if an application puts emphasis on the mass production, the multi-layer membrane device is an excellent choice

Investigating optimal cutting configurations for the contour method of weld residual stress measurement

The present work examines optimal cutting configurations for the measurement of weld residual stresses (WRS) using the contour method. The accuracy of a conventional, single-cut configuration that employs rigid clamping is compared with novel, double-embedded cutting configurations that rely on specimen self-constraint during cutting. Numerical analyses examine the redistribution of WRS and the development of cutting-induced plasticity (CIP) in a three-pass austenitic slot weld (NeT TG4) during the cutting procedure for each configuration. Stress intensity factor (SIF) analyses are first used as a screening tool; these analyses characterise lower stress intensities near the cutting surface when double-embedded cutting configurations are used, relative to SIF profiles from a single-cut process. The lower stress intensities indicate the development of CIP – which will ultimately affect back-calculated WRS – is less likely to occur when using an embedded configuration. The improvements observed for embedded cutting approaches are confirmed using three-dimensional finite element (FE) cutting simulations. The simulations reveal significant localised plasticity that forms in the material ligaments located between the pilot holes and the outer edges of the specimen. This plasticity is caused by WRS redistribution during the cutting process. The compressive plasticity in these material ligaments is shown to reduce the overall tensile WRS near the weld region before this region is sectioned, thereby significantly reducing the amount of CIP when cutting through the weld region at a later stage of the cutting procedure. Further improvements to the embedded cutting configuration are observed when the equilibrating compressive stresses in material ligaments are removed entirely (via sectioning) prior to sectioning of the high WRS region in the vicinity of the weld. All numerical results are validated against a series of WRS measurements performed using the contour method on a set of NeT TG4 benchmark weld specimens

Relaxations of mixed integer sets from lattice-free polyhedra

This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitel

Three dimensional passive localization for single path arrival with unknown starting conditions

Introduced in this paper is the time difference of arrival (TDoA) conic approximation method (TCAM), a technique for passive localization in three dimensions with unknown starting conditions. The TDoA of a mutually detected signal across pairs of detectors is used to calculate the relative angle between the signal source and the center point of the separation between the detectors in the pair. The relative angle is calculated from the TDoA using a mathematical model called the TDoA approximation of the zenith angle (TAZA). The TAZA angle defines the opening angle of a conic region of probability that contains the signal source, produced by each detector pair. The intersecting region of probability is determined from the conic regions of probability and represents the volumetric region with the highest probability of containing the signal source. TCAM was developed and tested using synthetic data in a simulated environment

Three dimensional passive localization for single path arrival with unknown starting conditions

Introduced in this paper is the time difference of arrival (TDoA) conic approximation method (TCAM), a technique for passive localization in three dimensions with unknown starting conditions. The TDoA of a mutually detected signal across pairs of detectors is used to calculate the relative angle between the signal source and the center point of the separation between the detectors in the pair. The relative angle is calculated from the TDoA using a mathematical model called the TDoA approximation of the zenith angle (TAZA). The TAZA angle defines the opening angle of a conic region of probability that contains the signal source, produced by each detector pair. The intersecting region of probability is determined from the conic regions of probability and represents the volumetric region with the highest probability of containing the signal source. TCAM was developed and tested using synthetic data in a simulated environment