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Graphical models are a popular approach to find dependence and conditional independence relationships between gene expressions. Directed acyclic graphs (DAGs) are a special class of directed graphical models, where all the edges are directed edges and contain no directed cycles. The DAGs are well known models for discovering causal relationships between genes in gene regulatory networks. However, estimating DAGs without assuming known ordering is challenging due to high dimensionality, the acyclic constraints, and the presence of equivalence class from observational data. To overcome these challenges, we propose a two-stage adaptive Lasso approach, called NS-DIST, which performs neighborhood selection (NS) in stage 1, and then estimates DAGs by the discrete improving search with Tabu (DIST) algorithm within the selected neighborhood. Simulation studies are presented to demonstrate the effectiveness of the method and its computational efficiency. Two real data examples are used to demonstrate the practical usage of our method for gene regulatory network inference. Supplementary materials for this article are available online.

We consider probabilistically constrained linear programs with general distributions for the uncertain parameters. These problems involve non-convex feasible sets. We develop a branch-and-bound algorithm that searches for a global optimal solution to this problem by successively partitioning the non-convex feasible region and by using bounds on the objective function to fathom inferior partition elements. This basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partition elements and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires solving linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented. [PUBLICATION ABSTRACT]

We study a deterministic maritime inventory routing problem with a long planning horizon. For instances with many ports and many vessels, mixed-integer linear programming (MIP) solvers often require hours to produce good solutions even when the planning horizon is 90 or 120 periods. Building on the recent successes of approximate dynamic programming (ADP) for road-based applications within the transportation community, we develop an ADP procedure to generate good solutions to these problems within minutes. Our algorithm operates by solving many small subproblems (one for each time period) and by collecting information about how to produce better solutions. Our main contribution to the ADP community is an algorithm that solves MIP subproblems and uses separable piecewise linear continuous, but not necessarily concave or convex, value function approximations and requires no off-line training. Our algorithm is one of the first of its kind for maritime transportation problems and represents a significant departure from the traditional methods used. In particular, whereas virtually all existing methods are \"MIP-centric,\" i.e., they rely heavily on a solver to tackle a nontrivial MIP to generate a good or improving solution in a couple of minutes, our framework puts the effort on finding suitable value function approximations and places much less responsibility on the solver. Computational results illustrate that with a relatively simple framework, our ADP approach is able to generate good solutions to instances with many ports and vessels much faster than a commercial solver emphasizing feasibility and a popular local search procedure.

In this paper, we examine a mixed integer linear programming reformulation for mixed integer bilinear problems where each bilinearterm involves the product of a nonnegative integer variable and a nonnegative continuous variable. This reformulation is obtained by first replacing a general integer variable with its binary expansion and then using McCormick envelopes to linearize the resulting product of continuous and binary variables. We present the convex hull of the underlying mixed integer linear set. The effectiveness of this reformulation and associated facet-defining inequalities are computationally evaluated on five classes of instances. [PUBLICATION ABSTRACT]

The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances.

This paper discusses an outer-approximation guided optimization method for constrained neural network inverse problems with rectified linear units. The constrained neural network inverse problems refer to an optimization problem to find the best set of input values of a given trained neural network in order to produce a predefined desired output in presence of constraints on input values. This paper analyzes the characteristics of optimal solutions of neural network inverse problems with rectified activation units and proposes an outer-approximation algorithm by exploiting their characteristics. The proposed outer-approximation guided optimization comprises primal and dual phases. The primal phase incorporates neighbor curvatures with neighbor outer-approximations to expedite the process. The dual phase identifies and utilizes the structure of local convex regions to improve the convergence to a local optimal solution. At last, computation experiments demonstrate the superiority of the proposed algorithm compared to a projected gradient method.

Monotonic optimization consists of minimizing or maximizing a monotonic objective function over a set of constraints defined by monotonic functions. Many optimization problems in economics and engineering often have monotonicity while lacking other useful properties, such as convexity. This thesis is concerned with the development and application of global optimization algorithms for monotonic optimization problems. First, we propose enhancements to an existing outer-approximation algorithm—called the Polyblock Algorithm—for monotonic optimization problems. The enhancements are shown to significantly improve the computational performance of the algorithm while retaining the convergence properties. Next, we develop a generic branch-and-bound algorithm for monotonic optimization problems. A computational study is carried out for comparing the performance of the Polyblock Algorithm and variants of the proposed branch-and-bound scheme on a family of separable polynomial programming problems. Finally, we study an important class of monotonic optimization problems—probabilistically constrained linear programs. We develop a branch-and-bound algorithm that searches for a global solution to the problem. The basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partitions and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires the solution of only linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented.

Objective This study aimed to develop and validate post- and preoperative models for predicting recurrence after curative-intent surgery using an FDG PET-CT metabolic parameter to improve the prognosis of patients with synchronous colorectal cancer liver metastasis (SCLM). Methods In this retrospective multicenter study, consecutive patients with resectable SCLM underwent upfront surgery between 2006 and 2015 (development cohort) and between 2006 and 2017 (validation cohort). In the development cohort, we developed and internally validated the post- and preoperative models using multivariable Cox regression with an FDG metabolic parameter (metastasis-to-primary-tumor uptake ratio [M/P ratio]) and clinicopathological variables as predictors. In the validation cohort, the models were externally validated for discrimination, calibration, and clinical usefulness. Model performance was compared with that of Fong’s clinical risk score (FCRS). Results A total of 374 patients (59.1 ± 10.5 years, 254 men) belonged in the development cohort and 151 (60.3 ± 12.0 years, 94 men) in the validation cohort. The M/P ratio and nine clinicopathological predictors were included in the models. Both postoperative and preoperative models showed significantly higher discrimination than FCRS ( p < .05) in the external validation (time-dependent AUC = 0.76 [95% CI 0.68–0.84] and 0.76 [0.68–0.84] vs. 0.65 [0.57–0.74], respectively). Calibration plots and decision curve analysis demonstrated that both models were well calibrated and clinically useful. The developed models are presented as a web-based calculator ( https://cpmodel.shinyapps.io/SCLM/ ) and nomograms. Conclusions FDG metabolic parameter-based prognostic models are well-calibrated recurrence prediction models with good discriminative power. They can be used for accurate risk stratification in patients with SCLM. Key Points • In this multicenter study, we developed and validated prediction models for recurrence in patients with resectable synchronous colorectal cancer liver metastasis using a metabolic parameter from FDG PET-CT. • The developed models showed good predictive performance on external validation, significantly exceeding that of a pre-existing model. • The models may be utilized for accurate patient risk stratification, thereby aiding in therapeutic decision-making.