Between steps: Intermediate relaxations between big-M and convex hull formulations

This work develops a class of relaxations in between the big-M and convex hull formulations of disjunctions, drawing advantages from both. The proposed "P-split" formulations split convex additively separable constraints into P partitions and form the convex hull of the partitioned disjuncts. Parameter P represents the trade-off of model size vs. relaxation strength. We examine the novel formulations and prove that, under certain assumptions, the relaxations form a hierarchy starting from a big-M equivalent and converging to the convex hull. We computationally compare the proposed formulations to big-M and convex hull formulations on a test set including: K-means clustering, P_ball problems, and ReLU neural networks. The computational results show that the intermediate P-split formulations can form strong outer approximations of the convex hull with fewer variables and constraints than the extended convex hull formulations, giving significant computational advantages over both the big-M and convex hull

LineWalker: Line Search for Black Box Derivative-Free Optimization and Surrogate Model Construction

This paper describes a simple, but effective sampling method for optimizing and learning a discrete approximation (or surrogate) of a multi-dimensional function along a one-dimensional line segment of interest. The method does not rely on derivative information and the function to be learned can be a computationally-expensive ``black box'' function that must be queried via simulation or other means. It is assumed that the underlying function is noise-free and smooth, although the algorithm can still be effective when the underlying function is piecewise smooth. The method constructs a smooth surrogate on a set of equally-spaced grid points by evaluating the true function at a sparse set of judiciously chosen grid points. At each iteration, the surrogate's non-tabu local minima and maxima are identified as candidates for sampling. Tabu search constructs are also used to promote diversification. If no non-tabu extrema are identified, a simple exploration step is taken by sampling the midpoint of the largest unexplored interval. The algorithm continues until a user-defined function evaluation limit is reached. Numerous examples are shown to illustrate the algorithm's efficacy and superiority relative to state-of-the-art methods, including Bayesian optimization and NOMAD, on primarily nonconvex test functions.Comment: 58 pages, 7 main figures, 29 total figure

Partition-based formulations for mixed-integer optimization of trained ReLU neural networks

This paper introduces a class of mixed-integer formulations for trained ReLU neural networks. The approach balances model size and tightness by partitioning node inputs into a number of groups and forming the convex hull over the partitions via disjunctive programming. At one extreme, one partition per input recovers the convex hull of a node, i.e., the tightest possible formulation for each node. For fewer partitions, we develop smaller relaxations that approximate the convex hull, and show that they outperform existing formulations. Specifically, we propose strategies for partitioning variables based on theoretical motivations and validate these strategies using extensive computational experiments. Furthermore, the proposed scheme complements known algorithmic approaches, e.g., optimization-based bound tightening captures dependencies within a partition

Proteomic maps of breast cancer subtypes

Systems-wide profiling of breast cancer has almost always entailed RNA and DNA analysis by microarray and sequencing techniques. Marked developments in proteomic technologies now enable very deep profiling of clinical samples, with high identification and quantification accuracy. We analysed 40 oestrogen receptor positive (luminal), Her2 positive and triple negative breast tumours and reached a quantitative depth of >10,000 proteins. These proteomic profiles identified functional differences between breast cancer subtypes, related to energy metabolism, cell growth, mRNA translation and cell-cell communication. Furthermore, we derived a signature of 19 proteins, which differ between the breast cancer subtypes, through support vector machine (SVM)-based classification and feature selection. Remarkably, only three proteins of the signature were associated with gene copy number variations and eleven were also reflected on the mRNA level. These breast cancer features revealed by our work provide novel insights that may ultimately translate to development of subtype-specific therapeutics

Method for solving generalized convex nonsmooth mixed-integer nonlinear programming problems

In this paper, we generalize the extended supporting hyperplane algorithm for a convex continuously differentiable mixed-integer nonlinear programming problem to solve a wider class of nonsmooth problems. The generalization is made by using the subgradients of the Clarke subdifferential instead of gradients. Consequently, all the functions in the problems are assumed to be locally Lipschitz continuous. The algorithm is shown to converge to a global minimum of an MINLP problem if the objective function is convex and the constraint functions are f degrees-pseudoconvex. With some additional assumptions, the constraint functions may be f degrees-quasiconvex

Proteomic maps of breast cancer subtypes

Systems-wide profiling of breast cancer has almost always entailed RNA and DNA analysis by microarray and sequencing techniques. Marked developments in proteomic technologies now enable very deep profiling of clinical samples, with high identification and quantification accuracy. We analysed 40 oestrogen receptor positive (luminal), Her2 positive and triple negative breast tumours and reached a quantitative depth of >10,000 proteins. These proteomic profiles identified functional differences between breast cancer subtypes, related to energy metabolism, cell growth, mRNA translation and cell-cell communication. Furthermore, we derived a signature of 19 proteins, which differ between the breast cancer subtypes, through support vector machine (SVM)-based classification and feature selection. Remarkably, only three proteins of the signature were associated with gene copy number variations and eleven were also reflected on the mRNA level. These breast cancer features revealed by our work provide novel insights that may ultimately translate to development of subtype-specific therapeutics.</p

Engineering Bispecificity into a Single Albumin-Binding Domain

Bispecific antibodies as well as non-immunoglobulin based bispecific affinity proteins are considered to have a very high potential in future biotherapeutic applications. In this study, we report on a novel approach for generation of extremely small bispecific proteins comprised of only a single structural domain. Binding to tumor necrosis factor-α (TNF-α) was engineered into an albumin-binding domain while still retaining the original affinity for albumin, resulting in a bispecific protein composed of merely 46 amino acids. By diversification of the non albumin-binding side of the three-helix bundle domain, followed by display of the resulting library on phage particles, bispecific single-domain proteins were isolated using selections with TNF-α as target. Moreover, based on the obtained sequences from the phage selection, a second-generation library was designed in order to further increase the affinity of the bispecific candidates. Staphylococcal surface display was employed for the affinity maturation, enabling efficient isolation of improved binders as well as multiparameter-based sortings with both TNF-α and albumin as targets in the same selection cycle. Isolated variants were sequenced and the binding to albumin and TNF-α was analyzed. This analysis revealed an affinity for TNF-α below 5 nM for the strongest binders. From the multiparameter sorting that simultaneously targeted TNF-α and albumin, several bispecific candidates were isolated with high affinity to both antigens, suggesting that cell display in combination with fluorescence activated cell sorting is a suitable technology for engineering of bispecificity. To our knowledge, the new binders represent the smallest engineered bispecific proteins reported so far. Possibilities and challenges as well as potential future applications of this novel strategy are discussed

Correlation of computed tomography with carotid plaque transcriptomes associates calcification with lesion-stabilization

Background and aims: Unstable carotid atherosclerosis causes stroke, but methods to identify patients and lesions at risk are lacking. We recently found enrichment of genes associated with calcification in carotid plaques from asymptomatic patients. Here, we hypothesized that calcification represents a stabilising feature of plaques and investigated how macro-calcification, as estimated by computed tomography (CT), correlates with gene expression profiles in lesions. Methods: Plaque calcification was measured in pre-operative CT angiographies. Plaques were sorted into high- and lowcalcified, profiled with microarrays, followed by bioinformatic analyses. Immunohistochemistry and qPCR were performed to evaluate the findings in plaques and arteries with medial calcification from chronic kidney disease patients. Results: Smooth muscle cell (SMC) markers were upregulated in high-calcified plaques and calcified plaques from symptomatic patients, whereas macrophage markers were downregulated. The most enriched processes in high-calcified plaques were related to SMCs and extracellular matrix (ECM) organization, while inflammation, lipid transport and chemokine signaling were repressed. These findings were confirmed in arteries with high medial calcification. Proteoglycan 4 (PRG4) was identified as the most upregulated gene in association with plaque calcification and found in the ECM, SMA+ and CD68+/TRAP + cells. Conclusions: Macro-calcification in carotid lesions correlated with a transcriptional profile typical for stable plaques, with altered SMC phenotype and ECM composition and repressed inflammation. PRG4, previously not described in atherosclerosis, was enriched in the calcified ECM and localized to activated macrophages and smooth muscle-like cells. This study strengthens the notion that assessment of calcification may aid evaluation of plaque phenotype and stroke risk.The European Union’s Horizon 2020/Marie Sklodowska-Curie grant agreement No 722609 (INTRICARE);Swedish Heart and Lung FoundationSwedish Research Council (K2009-65X-2233-01-3, K2013- 65X-06816-30-4, 349-2007-8703)Uppdrag Besegra Stroke (P581/ 2011-123)Stockholm County Council (ALF2011-0260, ALF-2011- 0279)Swedish Society for Medical ResearchTore Nilsson’s FoundationMagnus Bergvall’s FoundationKarolinska Institutet FoundationEuropean Commission (722609)Publishe