Stochastic geometric optimization with joint probabilistic constraints

Stochastic geometric programming with joint probabilistic constraints

We consider a supply chain operating in an uncertain environment: The customers’ demand is characterized by a discrete probability distribution. A probabilistic programming approach is adopted for constructing an inventory-production-distribution plan over a multiperiod planning horizon. The plan does not allow the backlogging of the unsatisfied demand, and minimizes the costs of the supply chain while enabling it to reach a prescribed nonstockout service level. It is a strategic plan that hedges against undesirable outcomes, and that can be adjusted to account for possible favorable realizations of uncertain quantities. A modular, integrated, and computationally tractable method is proposed for the solution of the associated stochastic mixed-integer optimization problems containing joint probabilistic constraints with dependent right-hand side variables. The concept of p-efficiency is used to construct a finite number of demand trajectories, which in turn are employed to solve problems with joint probabilistic constraints. We complement this idea by designing a preordered set-based preprocessing algorithm that selects a subset of promising p-efficient demand trajectories. Finally, to solve the resulting disjunctive mixed-integer programming problem, we implement a special column-generation algorithm that limits the risk of congestion in the resources of the supply chain. The methodology is validated on an industrial problem faced by a large chemical supply chain and turns out to be very efficient: it finds a solution with a minimal integrality gap and provides substantial cost savings.

Sequential equi-marginal optimization method for ranking strategies for thermal building renovation

A sequential optimization algorithm (SOA) for resource allocation in a cyclic-prefixed single-carrier cognitive relay system is proposed in this study. Both subcarrier pairing (SP) and power allocation are performed subject to a primary user interference constraint to minimize the mean squared error of frequency-domain equalization at the secondary destination receiver. Under uniform power allocation at the secondary source and optimal power allocation at the secondary relay, the ordered SP is proven to be asymptotically optimal in maximizing the matched filter bound on the signal-to-interference-plus-noise ratio. SOA implements the ordered SP before power allocation optimization by decoupling the ordered SP from the power allocation. Simulation results show that SOA can optimize resource allocation efficiently by significantly reducing complexity.

Many algorithms have been devised for minimizing the costs associated with obtaining the answer to a single, isolated query in a distributed database system. However, if more than one query may be processed by the system at the same time and if the arrival times of the queries are unknown, the determination of optimal query-processing strategies becomes a stochastic optimization problem. In order to cope with such problems, a theoretical state-transition model is presented that treats the system as one operating under a stochastic load. Query-processing strategies may then be distributed over the processors of a network as probability distributions, in a manner which accommodates many queries over time. It is then shown that the model leads to the determination of optimal query-processing strategies as the solution of mathematical programming problems, and analytical results for several examples are presented. Furthermore, a divide-and-conquer approach is introduced for decomposing stochastic query optimization problems into distinct subproblems for processing queries sequentially and in parallel.

The chemical complexity and diversity of an Athabasca asphaltene sample was described using a series of molecular representations. The molecular representations were created with a Monte Carlo construction method that represented molecules with a series of aromatic and aliphatic groups. After the groups were randomly sampled for a molecule, a connection algorithm linked them together to form molecules consisting of aromatic groups connected by aliphatic chains and thioethers. A sequential nonlinear optimization algorithm was used to select a small subset of molecules that were consistent with elemental, molecular weight, and NMR spectroscopy (both 13C and 1H) data. To accurately represent the analytical data for the asphaltene sample, a minimum of five molecules was needed. On the basis of the results of the sequential optimization, at least 50 molecules in the starting population were required to produce an analytically consistent molecular representation.

In this paper, we consider the convex stochastic conic optimization defined by expectations. An augmented Lagrangian stochastic approximation (ALSAcp) method is proposed to solve this convex stochastic conic optimization problem and the regrets of this method are analyzed. Under mild conditions, we show that this method exhibits ${\rm~O}(T^{-1/2})$ regret for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where $T$ denotes the number of iterations. Moreover, we show that, with at least $1-1/T$ probability, the method has no more than ${\rm~O}(\log~(T)/\sqrt{T})$ for both objective descent regret and constraint violation regret.

We study the stochastic versions of a broad class of combinatorial problems where the weights of the elements in the input dataset are uncertain. The class of problems that we study includes shortest paths, minimum weight spanning trees, and minimum weight matchings over probabilistic graphs, and other combinatorial problems like knapsack. We observe that the expected value is inadequate in capturing different types of risk averse or risk-prone behaviors, and instead we consider a more general objective which is to maximize the expected utility of the solution for some given utility function, rather than the expected weight (expected weight becomes a special case). We show that we can obtain a polynomial time approximation algorithm with additive error ϵ for any ϵ >; 0, if there is a pseudopolynomial time algorithm for the exact version of the problem (This is true for the problems mentioned above) and the maximum value of the utility function is bounded by a constant. Our result generalizes several prior results on stochastic shortest path, stochastic spanning tree, and stochastic knapsack. Our algorithm for utility maximization makes use of the separability of exponential utility and a technique to decompose a general utility function into exponential utility functions, which may be useful in other stochastic optimization problems.

Most of the existing literature concerning lifetime extension of battery energy storage (BES) devices mainly focus on designing controllers in BES devices themselves. In this paper, the problem of BES lifetime extension in an islanded microgrid is considered from a different perspective, in the sense that such target is achieved by adjusting output power of controllable power generation devices. To obtain accurate power models of photovoltaic panels and loads, a new hybrid modeling method utilizing both neural networks and stochastic differential equations is proposed. We formulate the problem of extending the service life of BES devices as a stochastic optimal control problem. Meanwhile, the situation of over-control in micro-turbines is effectively avoided. We solve such stochastic optimization problem using the dynamic programming solver provided by the open source C++ program BOCOPHJB. Finally, the obtained optimal controller is verified with simulations.

Contributors

Design optimization of multi-body systems is challenged by the highly non-linear and computationally expensive (for large-scale systems) nature of many multi-body responses. In this paper, a sequential approximate optimization method is presented in order to achieve accurate and efficient design optimization of multi-body systems under impact loads. The sequential approximate optimization method is obtained by coupling a conventional optimization algorithm with a statistical model building technique, response surface methodology (RSM). In this method, a multi-body optimization problem is approximated as a series of simpler subproblems, using RSM and a genetic algorithm, before it is solved by a numerical optimization algorithm. The approximate optimization method is applied to several multi-body impact problems. These include a linear impact absorber and lumped-mass systems representing simplified vehicle-barrier and vehicle-vehicle frontal impacts. The effect of the choice of various approximation methods is also addressed. The results demonstrate that the use of RSM coupled with conventional optimization methods leads to efficient and effective design tools for multi-body systems under impacts.

Sequential quadratic optimization algorithms are proposed for solving smooth nonlinear optimization problems with equality constraints. The main focus is an algorithm proposed for the case when the...

Chance-constrained model predictive control for drinking water networks

To address the challenging problem of measuring juvenile sleep deprivation, we test the hypothesis that a pupillographic method found successful for adult narcoleptics might also discriminate between sleep deprived juveniles acting as their own controls. A linear, nonstationary model relating pupillary diameter and a random photic stimulus are estimated by recursive regressions from pupillographic records of 8 juveniles of median age 7 years acting as their own rested controls. The estimated pupillary impulse response noise functions are stochastically optimized using the Kullback divergence measure to maximally separate the sleep deprived records from the control records. Both the average and covariance statistics of the estimated pupillary noise functions exhibit statistically significant differences between sleep deprived and rested subjects. The main result is that sleep deprivation decreases pupillary noise variance; a finding consistent with a previous study of adult narcoleptics. Further, it was found that virtually the same stochastic parameters were optimal for the juvenile sleep deprived data and for the previous adult narcoleptic study. Although our results are preliminary, the consistent reduction of pupillary noise appears to justify a comprehensive clinical trial across a broad range of age classes. In addition, the finding that the same parameters stochastically optimize both juvenile and adult recordings suggests the procedure holds promise as a clinical test which could produce sleep deprivation measures simultaneous with data collection.

Weighted-sum energy efficiency (WSEE) is a key performance metric in heterogeneous networks, where the nodes may have different energy efficiency requirements. Nevertheless, WSEE maximization is a challenging problem due to its nonconvex sum-of-ratios form. Unlike previous work, this letter presents a systematic approach to WSEE maximization under not only power constraints, but also data rate constraints, using a general signal-to-interference-plus-noise-ratio expression. In particular, the original problem is transformed into an equivalent form, and then a sequential convex optimization (SCO) algorithm is proposed. This algorithm is theoretically guaranteed to converge for any initial feasible point, and, under suitable constraint qualifications, achieves a Karush-Kuhn-Tucker (KKT) solution. Furthermore, we provide remarkable extensions to the proposed methodology, including systems with multiple resource blocks as well as a more general power consumption model which is not necessarily a convex function of the transmit powers. Finally, numerical analysis reveals that the proposed algorithm exhibits fast convergence, low complexity, and robustness (insensitivity to initial points).