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Alexander Mitsos - One of the best experts on this subject based on the ideXlab platform.

  • Linearization of McCormick relaxations and hybridization with the auxiliary variable method
    Journal of Global Optimization, 2021
    Co-Authors: Jaromił Najman, Dominik Bongartz, Alexander Mitsos
    Abstract:

    The computation of lower bounds via the solution of convex lower bounding problems depicts current state-of-the-art in deterministic global optimization. Typically, the nonlinear convex relaxations are further underestimated through linearizations of the convex underestimators at one or several points resulting in a lower bounding linear optimization problem. The selection of linearization points substantially affects the tightness of the lower bounding linear problem. Established methods for the computation of such linearization points, e.g., the sandwich algorithm, are already available for the auxiliary variable method used in state-of-the-art deterministic global optimization solvers. In contrast, no such methods have been proposed for the (multivariate) McCormick relaxations. The difficulty of determining a good set of linearization points for the McCormick technique lies in the fact that no auxiliary variables are introduced and thus, the linearization points have to be determined in the space of original optimization variables. We propose algorithms for the computation of linearization points for convex relaxations constructed via the (multivariate) McCormick theorems. We discuss alternative approaches based on an adaptation of Kelley’s algorithm; computation of all vertices of an n -simplex; a combination of the two; and random selection. All algorithms provide substantial speed ups when compared to the single point strategy used in our previous works. Moreover, we provide first results on the hybridization of the auxiliary variable method with the McCormick technique benefiting from the presented linearization strategies resulting in additional computational advantages.

  • On tightness and anchoring of McCormick and other relaxations
    Journal of Global Optimization, 2019
    Co-Authors: Jaromił Najman, Alexander Mitsos
    Abstract:

    We say that a convex relaxation of a function is anchored at a particular point in their domains if the values of the function and the relaxation at this point are equal. The opposite of anchoring is offset, i.e., a positive difference between the function and its convex relaxation values over the entire domain. We present theoretical results supported by theoretical and numerical examples showing that anchoring (at corner points) is a useful property but neither necessary nor sufficient for favorable Hausdorff and pointwise convergence order of a relaxation-based bounding scheme. Next, we investigate the tightness and convergence behavior of McCormick relaxations in specific cases. McCormick relaxations have favorable convergence orders, but a positive offset may still slow down the convergence within a simple branch-and-bound algorithm. We demonstrate that use of tighter underlying interval extensions can help reduce the offset and accelerate convergence.

  • Tighter McCormick relaxations through subgradient propagation
    Journal of Global Optimization, 2019
    Co-Authors: Jaromił Najman, Alexander Mitsos
    Abstract:

    Tight convex and concave relaxations are of high importance in deterministic global optimization. We present a method to tighten relaxations obtained by the McCormick technique. We use the McCormick subgradient propagation (Mitsos et al. in SIAM J Optim 20(2):573–601, 2009 ) to construct simple affine under- and overestimators of each factor of the original factorable function. Then, we minimize and maximize these affine relaxations in order to obtain possibly improved range bounds for every factor resulting in possibly tighter final McCormick relaxations. We discuss the method and its limitations, in particular the lack of guarantee for improvement. Subsequently, we provide numerical results for benchmark cases found in the MINLPLib2 library and case studies presented in previous works, where the McCormick technique appears to be advantageous, and discuss computational efficiency. We see that the presented algorithm provides a significant improvement in tightness and decrease in computational time, especially in the case studies using the reduced space formulation presented in (Bongartz and Mitsos in J Glob Optim 69:761–796, 2017 ).

  • tighter McCormick relaxations through subgradient propagation
    arXiv: Optimization and Control, 2017
    Co-Authors: Jaromil Najman, Alexander Mitsos
    Abstract:

    Tight convex and concave relaxations are of high importance in the field of deterministic global optimization. We present a heuristic to tighten relaxations obtained by the McCormick technique. We use the McCormick subgradient propagation (Mitsos et al., SIAM J. Optim., 2009) to construct simple affine under- and overestimators of each factor of the original factorable function. Then, we minimize and maximize these affine relaxations in order to obtain possibly improved range bounds for every factor resulting in possibly tighter final McCormick relaxations. We discuss the heuristic and its limitations, in particular the lack of guarantee for improvement. Subsequently, we provide numerical results for benchmark cases found in the COCONUT library and case studies presented in previous works and discuss computational efficiency. We see that the presented heuristic provides a significant improvement in tightness and decrease in computational time in many cases.

  • infeasible path global flowsheet optimization using McCormick relaxations
    Computer-aided chemical engineering, 2017
    Co-Authors: Dominik Bongartz, Alexander Mitsos
    Abstract:

    Abstract Deterministic global methods for flowsheet optimization have almost exclusively relied on equation-oriented formulations. The automatic propagation of McCormick relaxations and subgradients enables an alternative formulation similar to sequential modular infeasible path methods in local optimization that operate in a reduced space while moving most model variables and equations to external functions. The application of this reduced-space formulation is demonstrated for the Williams-Otto process. For suitable choices of tear streams and additional variables and equations left to the optimizer, it enables significant reductions in computational time compared to equation-oriented formulations.

Albino Bricolo - One of the best experts on this subject based on the ideXlab platform.

  • motor evoked potential monitoring improves outcome after surgery for intramedullary spinal cord tumors a historical control study
    Neurosurgery, 2006
    Co-Authors: Francesco Sala, Giorgio Palandri, Elisabetta Basso, P Lanteri, Vedran Deletis, F Faccioli, Albino Bricolo
    Abstract:

    OBJECTIVE: The value of intraoperative neurophysiological monitoring (INM) during intramedullary spinal cord tumor surgery remains debated. This historical control study tests the hypothesis that INM monitoring improves neurological outcome. METHODS: In 50 patients operated on after September 2000, we monitored somatosensory evoked potentials and transcranially elicited epidural (D-wave) and muscle motor evoked potentials (INM group). The historical control group consisted of 50 patients selected from among 301 patients who underwent intramedullary spinal cord tumor surgery, previously operated on by the same team without INM. Matching by preoperative neurological status (McCormick scale), histological findings, tumor location, and extent of removal were blind to outcome. A more than 50% somatosensory evoked potential amplitude decrement influenced only myelotomy. Muscle motor evoked potential disappearance modified surgery, but more than 50% D-wave amplitude decrement was the major indication to stop surgery. The postoperative to preoperative McCormick grade variation at discharge and at a follow-up of at least 3 months was compared between the two groups (Student's t tests). RESULTS: Follow-up McCormick grade variation in the INM group (mean, +0.28) was significantly better (P = 0.0016) than that of the historical control group (mean, -0.16). At discharge, there was a trend (P = 0.1224) toward better McCormick grade variation in the INM group (mean, -0.26) than in the historical control group (mean, -0.5). CONCLUSION: The applied motor evoked potential methods seem to improve long-term motor outcome significantly. Early motor outcome is similar because of transient motor deficits in the INM group, which can be predicted at the end of surgery by the neurophysiological profile of patients.

Francois Porchet - One of the best experts on this subject based on the ideXlab platform.

  • assessment of outcome in patients undergoing surgery for intradural spinal tumor using the multidimensional patient rated core outcome measures index and the modified McCormick scale
    Neurosurgical Focus, 2015
    Co-Authors: David Bellut, Jankarl Burkhardt, Anne F Mannion, Francois Porchet
    Abstract:

    OBJECT The aim of this study was to evaluate outcome in patients undergoing surgical treatment for intradural spinal tumor using a patient-oriented, self-rated, outcome instrument and a physician-based disease-specific instrument. METHODS Prospectively collected data from 63 patients with intradural spinal tumor were analyzed in relation to scores on the multidimensional patient-rated Core Outcome Measures Index (COMI) and the physician-rated modified McCormick Scale, before and at 3 and 12 months after surgery. RESULTS There was no statistically significant difference between the scores on the modified McCormick Scale preoperatively and at the 3-month follow-up, though there was a trend for improvement (p = 0.073); however, comparisons between the scores determined preoperatively and at the 12-month follow-up, as well as 3- versus 12-month follow-ups, showed a statistically significant improvement in each case (p 0.05) up to 12 months postoperatively. In contrast, the overall COMI score, "worst pain," quality of life, and social disability not only showed a significant reduction from before surgery to 3 months after surgery (p 0.05), but did show a significant improvement (p = 0.011) from 3 months to 12 months after surgery. At the 3- and 12-month follow-ups, 85.2% and 83.9% of patients, respectively, declared that the surgical procedure had helped/helped a lot; 95.1% and 95.2%, respectively, declared that they were satisfied/very satisfied with their care. CONCLUSIONS COMI is a feasible tool to use in the evaluation of baseline symptoms and outcome in patients undergoing surgery for intradural spinal tumor. COMI was able to detect changes in outcome at 3 months after surgery (before changes were apparent on the modified McCormick Scale) and on later postoperative follow-up. The COMI subdomains are valuable for monitoring the patient's reintegration into society and the work environment. The addition of an item that specifically covers neurological deficits may further increase the value of COMI in patients with spinal tumors.

Francesco Sala - One of the best experts on this subject based on the ideXlab platform.

  • motor evoked potential monitoring improves outcome after surgery for intramedullary spinal cord tumors a historical control study
    Neurosurgery, 2006
    Co-Authors: Francesco Sala, Giorgio Palandri, Elisabetta Basso, P Lanteri, Vedran Deletis, F Faccioli, Albino Bricolo
    Abstract:

    OBJECTIVE: The value of intraoperative neurophysiological monitoring (INM) during intramedullary spinal cord tumor surgery remains debated. This historical control study tests the hypothesis that INM monitoring improves neurological outcome. METHODS: In 50 patients operated on after September 2000, we monitored somatosensory evoked potentials and transcranially elicited epidural (D-wave) and muscle motor evoked potentials (INM group). The historical control group consisted of 50 patients selected from among 301 patients who underwent intramedullary spinal cord tumor surgery, previously operated on by the same team without INM. Matching by preoperative neurological status (McCormick scale), histological findings, tumor location, and extent of removal were blind to outcome. A more than 50% somatosensory evoked potential amplitude decrement influenced only myelotomy. Muscle motor evoked potential disappearance modified surgery, but more than 50% D-wave amplitude decrement was the major indication to stop surgery. The postoperative to preoperative McCormick grade variation at discharge and at a follow-up of at least 3 months was compared between the two groups (Student's t tests). RESULTS: Follow-up McCormick grade variation in the INM group (mean, +0.28) was significantly better (P = 0.0016) than that of the historical control group (mean, -0.16). At discharge, there was a trend (P = 0.1224) toward better McCormick grade variation in the INM group (mean, -0.26) than in the historical control group (mean, -0.5). CONCLUSION: The applied motor evoked potential methods seem to improve long-term motor outcome significantly. Early motor outcome is similar because of transient motor deficits in the INM group, which can be predicted at the end of surgery by the neurophysiological profile of patients.

Jon Lee - One of the best experts on this subject based on the ideXlab platform.

  • on branching point selection for trilinear monomials in spatial branch and bound the hull relaxation
    arXiv: Optimization and Control, 2017
    Co-Authors: Emily Speakman, Jon Lee
    Abstract:

    In Speakman and Lee (2017), we analytically developed the idea of using volume as a measure for comparing relaxations in the context of spatial branch-and-bound. Specifically, for trilinear monomials, we analytically compared the three possible "double-McCormick relaxations" with the tight convex-hull relaxation. Here, again using volume as a measure, for the convex-hull relaxation of trilinear monomials, we establish simple rules for determining the optimal branching variable and optimal branching point. Additionally, we compare our results with current software practice.

  • quantifying double McCormick
    Mathematics of Operations Research, 2017
    Co-Authors: Emily Speakman, Jon Lee
    Abstract:

    When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the bounds defining the box. In this manner, we precisely quantify the strength of the different possible relaxations defined by all three groupings, in addition to the trilinear hull itself. As a by-product, we characterize the best double-McCormick relaxation. We wish to emphasize that, in the context of spatial branch and bound for factorable formulations, our results do not only apply to variables in the input formulation. Our results apply to monomials that involve auxiliary variables as well. So, our results apply to the product of any three (possibly complicated) expressions in a formulation.

  • experimental validation of volume based comparison for double McCormick relaxations
    Integration of AI and OR Techniques in Constraint Programming, 2017
    Co-Authors: Emily Speakman, Jon Lee
    Abstract:

    Volume is a natural geometric measure for comparing polyhedral relaxations of non-convex sets. Speakman and Lee gave volume formulae for comparing relaxations of trilinear monomials, quantifying the strength of various natural relaxations. Their work was motivated by the spatial branch-and-bound algorithm for factorable mathematical-programming formulations. They mathematically analyzed an important choice that needs to be made whenever three or more terms are multiplied in a formulation. We experimentally substantiate the relevance of their main results to the practice of global optimization, by applying it to different relaxations of difficult box cubic problems (boxcup). In doing so, we find that, using their volume formulae, we can accurately predict the quality of a relaxation for boxcups based on the (box) parameters defining the feasible region. Specifically, we are able to conclude from our experiments that all of the relevant relaxations are round enough so that average objective gaps for a pair of boxcup relaxations can be predicted by appropriately combining the volumes of relaxations of the individual trilinear monomials.

  • experimental validation of volume based comparison for double McCormick relaxations
    arXiv: Optimization and Control, 2016
    Co-Authors: Emily Speakman, Jon Lee
    Abstract:

    Volume is a natural geometric measure for comparing polyhedral relaxations of non-convex sets. Speakman and Lee gave volume formulae for comparing relaxations of trilinear monomials, quantifying the strength of various natural relaxations. Their work was motivated by the spatial branch-and-bound algorithm for factorable mathematical-programming formulations. They mathematically analyzed an important choice that needs to be made whenever three or more terms are multiplied in a formulation. We experimentally substantiate the relevance of their main results to the practice of global optimization, by applying it to difficult box cubic problems (boxcup). In doing so, we find that, using their volume formulae, we can accurately predict the quality of a relaxation for boxcups based on the (box) parameters defining the feasible region.