Solving Mathematical Programs with Equilibrium Constraints as Nonlinear Programming: A New Framework

We present a new framework for the solution of mathematical programs with equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is viewed as a concentration of an unconstrained optimization which minimizes the complementarity measure and a nonlinear programming with general constraints. A strategy generalizing ideas of Byrd-Omojokun's trust region method is used to compute steps. By penalizing the tangential constraints into the objective function, we circumvent the problem of not satisfying MFCQ. A trust-funnel-like strategy is used to balance the improvements on feasibility and optimality. We show that, under MPEC-MFCQ, if the algorithm does not terminate in finite steps, then at least one accumulation point of the iterates sequence is an S-stationary point

A Sequential Quadratic Programming Method for Optimization with Stochastic Objective Functions, Deterministic Inequality Constraints and Robust Subproblems

In this paper, a robust sequential quadratic programming method of [1] for constrained optimization is generalized to problem with stochastic objective function, deterministic equality and inequality constraints. A stochastic line search scheme in [2] is employed to globalize the steps. We show that in the case where the algorithm fails to terminate in finite number of iterations, the sequence of iterates will converge almost surely to a Karush-Kuhn-Tucker point under the assumption of extended Mangasarian-Fromowitz constraint qualification. We also show that, with a specific sampling method, the probability of the penalty parameter approaching infinity is 0. Encouraging numerical results are reported

Nomogram for Predicting Long-Term Survival after Synchronous Resection for Hepatocellular Carcinoma and Inferior Vena Cava Tumor Thrombosis: A Multicenter Retrospective Study

Background. Although surgery for hepatocellular carcinoma (HCC) complicated with inferior vena cava tumor thrombus (IVCTT) may improve survival for some patients, prognostic markers remain elusive because of its rarity. We constructed a prognostic nomogram which predicts individualized survival benefit of curative-intent surgery for HCC patients with IVCTT. Methods. According to abdominothoracic anatomy of inferior vena cava (IVC), IVCTT can be divided into 3 types: inferior diaphragmic (ID), superior diaphragmic (SD), and intracardiac type (IC). Data of 64 HCC patients with IVCTT who underwent curative-intent surgery between 2008 and 2015 in four centers in China were analyzed retrospectively. Univariate and multivariate Cox regression analyses were conducted to select variables for the construction of a prognostic nomogram. Predictive accuracy and discriminative ability were examined by concordance index (C-index) and calibration curve. Results. Of 64 patients in the IVCTT classification, 37 (57.8%) were classified as ID type, 15 (23.4%) as SD type, and 12 (18.8%) as IC type. The 1-, 2-, 3-, and 5-year disease-specific survival (DSS) rates for patients in ID, SD, and IC groups were 94.4%, 55.6%, 71.4%, and 30.0%; 27.8%, 21.4%, 7.1%, and 0%; and 8.3%, 0%, 0%, and 0%, respectively. Independent factors included in the nomogram were ECOG performance status, AFP level ≥ 400 μg/L, tumor size ≥ 10 cm, portal vein tumor thrombosis, and IVCTT classification. The C-index of the nomogram was 0.812 (95% CI 0.761–0.873). The calibration plot for DSS probability showed excellent agreement between the prediction by nomogram and actual observation. Conclusions. Curative-intent surgery should be carefully evaluated and suggested according to our novel IVCTT classification. We have developed a visual web-based nomogram model to predict oncological prognosis of curative-intent surgery for HCC patients with IVCTT

Turing patterns with high-resolution formed without chemical reaction in thin-film solution of organic semiconductors.

Regular patterns can form spontaneously in chemical reaction-diffusion systems under non-equilibrium conditions as proposed by Alan Turing. Here, we found that regular patterns can be generated in uphill-diffusion solution systems without a chemical reaction process through both in-situ and ex-situ observations. Organic semiconductor solution is confined between two parallel plates with controlled micron/submicron-meter distance to minimize convection of the liquid and avoid spinodal precipitation at equilibrium. The solvent evaporation concentrates the solution gradually into an oversaturated non-equilibrium condition, under which a phase-transition occurs and ordered concentration-waves are generated. By proper tuning of the experimental parameter, multiple regular patterns with micro/nano-meter scaled features (line, square-grid, zig-zag, and fence-like patterns etc.) were observed. We explain the observed phenomenon as Turing-pattern generation resulted from uphill-diffusion and solution oversaturation. The generated patterns in the solutions can be condensed onto substrates to form structured micro/nanomaterials. We have fabricated organic semiconductor devices with such patterned materials to demonstrate the potential applications. Our observation may serve as a milestone in the progress towards a fundamental understanding of pattern formation in nature, like in biosystem, and pave a new avenue in developing self-assembling techniques of micro/nano structured materials