On Solving Convex Optimization Problems with Linear Ascending Constraints

In this paper, we propose two algorithms for solving convex optimization problems with linear ascending constraints. When the objective function is separable, we propose a dual method which terminates in a finite number of iterations. In particular, the worst case complexity of our dual method improves over the best-known result for this problem in Padakandla and Sundaresan [SIAM J. Optimization, 20 (2009), pp. 1185-1204]. We then propose a gradient projection method to solve a more general class of problems in which the objective function is not necessarily separable. Numerical experiments show that both our algorithms work well in test problems.Comment: 20 pages. The final version of this paper is published in Optimization Letter

Convex separable problems with linear and box constraints

In this work, we focus on separable convex optimization problems with linear and box constraints and compute the solution in closed-form as a function of some Lagrange multipliers that can be easily computed in a finite number of iterations. This allows us to bridge the gap between a wide family of power allocation problems of practical interest in signal processing and communications and their efficient implementation in practice.Comment: 5 pages, 2 figures. Published at IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2014

A Decomposition Algorithm for Nested Resource Allocation Problems

We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict convexity or differentiability is needed. The method solves a hierarchy of resource allocation subproblems, whose solutions are used to convert constraints on sums of resources into bounds for separate variables at higher levels. The resulting time complexity for the integer problem is $O(n \log m \log (B/n))$, and the complexity of obtaining an $\epsilon$-approximate solution for the continuous case is $O(n \log m \log (B/\epsilon))$, $n$ being the number of variables, $m$ the number of ascending constraints (such that $m < n$), $\epsilon$ a desired precision, and $B$ the total resource. This algorithm attains the best-known complexity when $m = n$, and improves it when $\log m = o(\log n)$. Extensive experimental analyses are conducted with four recent algorithms on various continuous problems issued from theory and practice. The proposed method achieves a higher performance than previous algorithms, addressing all problems with up to one million variables in less than one minute on a modern computer.Comment: Working Paper -- MIT, 23 page

Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded

Decision trees usefully represent sparse, high dimensional and noisy data. Having learned a function from this data, we may want to thereafter integrate the function into a larger decision-making problem, e.g., for picking the best chemical process catalyst. We study a large-scale, industrially-relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pre-trained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models, or they may wish to optimize a discrete model that particularly well-represents a data set. We develop several heuristic methods to find feasible solutions, and an exact, branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on concrete mixture design instance and a chemical catalysis industrial instance