A Two-Level Approach to Large Mixed-Integer Programs with Application to Cogeneration in Energy-Efficient Buildings

We study a two-stage mixed-integer linear program (MILP) with more than 1 million binary variables in the second stage. We develop a two-level approach by constructing a semi-coarse model (coarsened with respect to variables) and a coarse model (coarsened with respect to both variables and constraints). We coarsen binary variables by selecting a small number of pre-specified daily on/off profiles. We aggregate constraints by partitioning them into groups and summing over each group. With an appropriate choice of coarsened profiles, the semi-coarse model is guaranteed to find a feasible solution of the original problem and hence provides an upper bound on the optimal solution. We show that solving a sequence of coarse models converges to the same upper bound with proven finite steps. This is achieved by adding violated constraints to coarse models until all constraints in the semi-coarse model are satisfied. We demonstrate the effectiveness of our approach in cogeneration for buildings. The coarsened models allow us to obtain good approximate solutions at a fraction of the time required by solving the original problem. Extensive numerical experiments show that the two-level approach scales to large problems that are beyond the capacity of state-of-the-art commercial MILP solvers

An Optimal Control Model of Technology Transition

This paper discusses the use of optimization software to solve an optimal control problem arising in the modeling of technology transition. We set up a series of increasingly complex models with such features as learning-by-doing, adjustment cost, and capital investment. The models are written in continuous time and then discretized by using different methods to transform them into large-scale nonlinear programs. We use a modeling language and numerical optimization methods to solve the optimization problem. Our results are consistent with ndings in the literature and highlight the impact the discretization choice has on the solution and accuracy.

Portfolio selection models: A review and new directions

Modern Portfolio Theory (MPT) is based upon the classical Markowitz model which uses variance as a risk measure. A generalization of this approach leads to mean-risk models, in which a return distribution is characterized by the expected value of return (desired to be large) and a risk value (desired to be kept small). Portfolio choice is made by solving an optimization problem, in which the portfolio risk is minimized and a desired level of expected return is specified as a constraint. The need to penalize different undesirable aspects of the return distribution led to the proposal of alternative risk measures, notably those penalizing only the downside part (adverse) and not the upside (potential). The downside risk considerations constitute the basis of the Post Modern Portfolio Theory (PMPT). Examples of such risk measures are lower partial moments, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We revisit these risk measures and the resulting mean-risk models. We discuss alternative models for portfolio selection, their choice criteria and the evolution of MPT to PMPT which incorporates: utility maximization and stochastic dominance

Mean-risk models using two risk measures: A multi-objective approach

This paper proposes a model for portfolio optimisation, in which distributions are characterised and compared on the basis of three statistics: the expected value, the variance and the CVaR at a specified confidence level. The problem is multi-objective and transformed into a single objective problem in which variance is minimised while constraints are imposed on the expected value and CVaR. In the case of discrete random variables, the problem is a quadratic program. The mean-variance (mean-CVaR) efficient solutions that are not dominated with respect to CVaR (variance) are particular efficient solutions of the proposed model. In addition, the model has efficient solutions that are discarded by both mean-variance and mean-CVaR models, although they may improve the return distribution. The model is tested on real data drawn from the FTSE 100 index. An analysis of the return distribution of the chosen portfolios is presented

Unifying nonlinearly constrained nonconvex optimization

Derivative-based iterative methods for nonlinearly constrained nonconvex optimization usually share common algorithmic components, such as strategies for computing a descent direction and mechanisms that promote global convergence. Based on this observation, we introduce an abstract framework based on four common ingredients that describes most derivative-based iterative methods and unifies their workflows. We then present Uno, a modular C++ solver that implements our abstract framework and allows the automatic generation of various strategy combinations with no programming effort from the user. Uno is meant to (1) organize mathematical optimization strategies into a coherent hierarchy; (2) offer a wide range of efficient and robust methods that can be compared for a given instance; (3) enable researchers to experiment with novel optimization strategies; and (4) reduce the cost of development and maintenance of multiple optimization solvers. Uno's software design allows user to compose new customized solvers for emerging optimization areas such as robust optimization or optimization problems with complementarity constraints, while building on reliable nonlinear optimization techniques. We demonstrate that Uno is highly competitive against state-of-the-art solvers filterSQP, IPOPT, SNOPT, MINOS, LANCELOT, LOQO, and CONOPT on a subset of 429 small problems from the CUTEst collection. Uno is available as open-source software under the MIT license at https://github.com/cvanaret/Uno .Comment: Submitted to Mathematical Programming Computation journa

Sequential Linear Integer Programming for Integer Optimal Control with Total Variation Regularization

We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty allows us to prove the existence of minimizers of the integer optimal control problem. We introduce a local optimality concept for the problem, which arises from the infinite-dimensional perspective. In the case of a one-dimensional domain of the control function, we prove convergence of the iterates produced by our algorithm to points that satisfy first-order stationarity conditions for local optimality. We demonstrate the theoretical findings on a computational example

State elimination for mixed-integer optimal control of partial differential equations by semigroup theory

Mixed-integer optimal control problems governed by partial differential equations (MIPDECOs) are powerful modeling tools but also challenging in terms of theory and computation. We propose a highly efficient state elimination approach for MIPDECOs that are governed by partial differential equations that have the structure of an abstract ordinary differential equation in function space. This allows us to avoid repeated calculations of the states for all time steps, and our approach is applied only once before starting the optimization. The presentation of theoretical results is complemented by numerical experiments