Abstract. Theory and implementation for the global optimization of a wide class of algorithms is presented via convex/affine relaxations. The basis for the proposed relaxations is the systematic construction of subgradients for the convex relaxations of factorable functions by McCormick [Math. Prog., 10 (1976), pp. 147-175]. Similar to the convex relaxation, the subgradient propagation relies on the recursive application of a few rules, namely, the calculation of subgradients for addition, multiplication, and composition operations. Subgradients at interior points can be calculated for any factorable function for which a McCormick relaxation exists, provided that subgradients are known for the relaxations of the univariate intrinsic functions. For boundary points, additional assumptions are necessary. An automated implementation based on operator overloading is presented, and the calculation of bounds based on affine relaxation is demonstrated for illustrative examples. Two numerical examples for the global optimization of algorithms are presented. In both examples a parameter estimation problem with embedded differential equations is considered. The solution of the differential equations is approximated by algorithms with a fixed number of iterations. 1. Introduction. The development of deterministic algorithms based on continuous and/or discrete branch-and-bound [10,17,18] has facilitated the global optimization of nonconvex programs. The basic principle of branch-and-bound, and related algorithms such as branch-and-cut [19] and branch-and-reduce [27], is to bound the optimal objective value between a lower bound and an upper bound. By branching on the host set, these bounds become tighter and eventually converge. For minimization, upper bounds are typically obtained via a feasible point or via a local solution of the original program. For the lower bound, typically a convex or affine relaxation of the nonconvex program is constructed and solved to global optimality via a convex solver. Convex and concave envelopes or tight relaxations are known for a variety of simple nonlinear terms [1,33,35], and this allows the construction of convex and concave relaxations for a quite general class of functions through several methods [21,2,33,12]. Simple lower bounds from interval analysis are also widely used in global optimization, e.g., [6,7,25]. Such bounds are often weaker but less computationally expensive to evaluate than relaxation-based bounds. For instance, for a box-constrained problem, no linear program (LP) or convex nonlinear program (NLP) needs to be solved.The majority of the literature on global optimization considers nonconvex programs for which explicit functions are known for the objective and constraints. A more

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Understanding the mechanisms of cell function and drug action is a major endeavor in the pharmaceutical industry. Drug effects are governed by the intrinsic properties of the drug (i.e., selectivity and potency) and the specific signaling transduction network of the host (i.e., normal vs. diseased cells). Here, we describe an unbiased, phosphoproteomic-based approach to identify drug effects by monitoring drug-induced topology alterations. With our proposed method, drug effects are investigated under diverse stimulations of the signaling network. Starting with a generic pathway made of logical gates, we build a cell-type specific map by constraining it to fit 13 key phopshoprotein signals under 55 experimental conditions. Fitting is performed via an Integer Linear Program (ILP) formulation and solution by standard ILP solvers; a procedure that drastically outperforms previous fitting schemes. Then, knowing the cell's topology, we monitor the same key phosphoprotein signals under the presence of drug and we re-optimize the specific map to reveal drug-induced topology alterations. To prove our case, we make a topology for the hepatocytic cell-line HepG2 and we evaluate the effects of 4 drugs: 3 selective inhibitors for the Epidermal Growth Factor Receptor (EGFR) and a non-selective drug. We confirm effects easily predictable from the drugs' main target (i.e., EGFR inhibitors blocks the EGFR pathway) but we also uncover unanticipated effects due to either drug promiscuity or the cell's specific topology. An interesting finding is that the selective EGFR inhibitor Gefitinib inhibits signaling downstream the Interleukin-1alpha (IL1α) pathway; an effect that cannot be extracted from binding affinity-based approaches. Our method represents an unbiased approach to identify drug effects on small to medium size pathways which is scalable to larger topologies with any type of signaling interventions (small molecules, RNAi, etc). The method can reveal drug effects on pathways, the cornerstone for identifying mechanisms of drug's efficacy.

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