Convex Integer Optimization by Constantly Many Linear Counterparts

In this article we study convex integer maximization problems with composite objective functions of the form $f(Wx)$, where $f$ is a convex function on $\R^d$ and $W$ is a $d\times n$ matrix with small or binary entries, over finite sets $S\subset \Z^n$ of integer points presented by an oracle or by linear inequalities. Continuing the line of research advanced by Uri Rothblum and his colleagues on edge-directions, we introduce here the notion of {\em edge complexity} of $S$, and use it to establish polynomial and constant upper bounds on the number of vertices of the projection \conv(WS) and on the number of linear optimization counterparts needed to solve the above convex problem. Two typical consequences are the following. First, for any $d$, there is a constant $m(d)$ such that the maximum number of vertices of the projection of any matroid $S\subset\{0,1\}^n$ by any binary $d\times n$ matrix $W$ is $m(d)$ regardless of $n$ and $S$; and the convex matroid problem reduces to $m(d)$ greedily solvable linear counterparts. In particular, $m(2)=8$. Second, for any $d,l,m$, there is a constant $t(d;l,m)$ such that the maximum number of vertices of the projection of any three-index $l\times m\times n$ transportation polytope for any $n$ by any binary $d\times(l\times m\times n)$ matrix $W$ is $t(d;l,m)$; and the convex three-index transportation problem reduces to $t(d;l,m)$ linear counterparts solvable in polynomial time

Predictive Markers of Response to Neoadjuvant Durvalumab with Nab-Paclitaxel and Dose-Dense Doxorubicin/Cyclophosphamide in Basal-Like Triple-Negative Breast Cancer.

PurposeWe examined gene expression, germline variant, and somatic mutation features associated with pathologic response to neoadjuvant durvalumab plus chemotherapy in basal-like triple-negative breast cancer (bTNBC).Experimental designGermline and somatic whole-exome DNA and RNA sequencing, programmed death ligand 1 (PD-L1) IHC, and stromal tumor-infiltrating lymphocyte scoring were performed on 57 patients. We validated our results using 162 patients from the GeparNuevo randomized trial.ResultsGene set enrichment analysis showed that pathways involved in immunity (adaptive, humoral, innate), JAK-STAT signaling, cancer drivers, cell cycle, apoptosis, and DNA repair were enriched in cases with pathologic complete response (pCR), whereas epithelial-mesenchymal transition, extracellular matrix, and TGFβ pathways were enriched in cases with residual disease (RD). Immune-rich bTNBC with RD was enriched in CCL-3, -4, -5, -8, -23, CXCL-1, -3, -6, -10, and IL1, -23, -27, -34, and had higher expression of macrophage markers compared with immune-rich cancers with pCR that were enriched in IFNγ, IL2, -12, -21, chemokines CXCL-9, -13, CXCR5, and activated T- and B-cell markers (GZMB, CD79A). In the validation cohort, an immune-rich five-gene signature showed higher expression in pCR cases in the durvalumab arm (P = 0.040) but not in the placebo arm (P = 0.923) or in immune-poor cancers. Independent of immune markers, tumor mutation burden was higher, and PI3K, DNA damage repair, MAPK, and WNT/β-catenin signaling pathways were enriched in germline and somatic mutations in cases with pCR.ConclusionsThe TGFβ pathway is associated with immune-poor phenotype and RD in bTNBC. Among immune-rich bTNBC RD, macrophage/neutrophil chemoattractants dominate the cytokine milieu, and IFNγ and activated B cells and T cells dominate immune-rich cancers with pCR