A solution algorithm for non-convex mixed integer optimization problems with only few continuous variables

Low power wireless sensors, personal wireless hub (PWH) and receivers can reduce the workload of the paramedic staff in a hospital. In this paper, we use multiple PWHs to transfer sensor data to the main central controller, which helps the wireless sensor devices. A well designed multiple PWH assignment and power control scheme can reduce the electromagnetic and in-band interference induced to the other medical devices in the hospital. We propose a framework and low complexity algorithm for interference aware joint power control and multiple PWH assignment (IAJPCPA) in a hospital building with cognitive radio capability. The proposed IAJPCPA is a non-convex mixed integer non-linear optimization problem (NC-MINLP) which is generally NP-hard. We present an efficient PWH assignment and power control scheme for IAJPCPA. We also propose an upper bound on the IAJPCPA that converts the non-convex problem into a convex optimization problem. We examine the effect of different system parameters.

We consider a wireless sensor network (WSN) tasked with tracking a process using a set of distributed nodes. Here multiple remote sensor nodes estimate the physical process (viz., a moving object) and transmit quantized estimates to a fusion center for processing. At the fusion node a BLUE (best linear unbiased estimation) approach is used to combine the sensor estimates and to create a final estimate of the state. In this framework the uncertainty of the overall estimate is derived and shown to depend on the individual sensor transmit energy and quantization levels, as well as the Kalman tracker uncertainty at the node. Since power and bandwidth are critically constrained resources in battery operated sensor nodes, we attempt to quantify the trade-off between the lifetime of the network and the estimation quality over time. Three different convex formulations of the underlying nonconvex mixed integer nonlinear optimization problem are presented. Unlike previous work this effort incorporates the operating state of the nodes into the decisions of the optimum bits and the transmission power levels based on a heuristic. Simulation results for all formulations demonstrate the quality of the state estimate as well as the extended lifetime of the WSN.

A neurodynamic optimization technique based on overestimator and underestimator functions for solving a class of non-convex optimization problems

In this paper, the classical exact absolute value penalty function method is used for solving a new class of nonconvex nonsmooth optimization problems. Nonconvex nondifferentiable optimization problems with both inequality and equality constraints are considered here, in which not all functions constituting them have the fundamental property of convex functions and most classes of generalized convex functions—namely, a stationary point of such a function is its global minimum. It is proved for such nonconvex optimization problems that there exists a finite threshold of penalty parameters equal to the largest absolute value of a Lagrange multiplier such that, for every penalty parameter exceeding this lower bound, there is the equivalence between an optimal solution in the original constrained minimization problem with ‐invex functions and a minimizer in its associated penalized optimization problem with the exact l1 penalty function. Further, under nondifferentiable ‐invexity assumptions, a characterization of a saddle point of the Lagrange function, defined for the considered constrained optimization problem in terms of minimizers of its associated exact penalized optimization problem with the exact l1 penalty function, is presented.

We develop new fast and efficient algorithms for designing single or multiple unimodular waveforms with good auto- and cross-correlation or weighted correlation properties, which are highly desired in radar and communication systems. The waveform design is based on the minimization of the integrated sidelobe level (ISL) and weighted ISL (WISL) of waveforms. As the corresponding problems can quickly grow to a large scale with increasing the code length and the number of waveforms, the main issue turns to be the development of fast large-scale optimization techniques. The difficulty is also that the corresponding optimization problems are nonconvex, but the required accuracy is high. Therefore, we formulate the ISL and WISL minimization problems as nonconvex quartic optimization problems in frequency domain, and then simplify them into quadratic problems via majorization-minimization technique, which is one of the basic techniques for addressing large-scale and/or nonconvex optimization problems. While designing our fast algorithms, we explore and use the inherent algebraic structures in objective functions to rewrite them into quartic forms, and in the case of WISL minimization, to derive additionally an alternative quartic form that allows us to apply the quartic-quadratic transformation. Our algorithms are applicable to large-scale unimodular waveform design problems as they are proved to have lower or comparable computational burden (analyzed theoretically) and faster convergence speed (confirmed by comprehensive simulations) than the state-of-the-art algorithms. In addition, the waveforms designed by our algorithms demonstrate better correlation properties compared to their counterparts.

Trajectory optimization for asteroid landing with two-phase free final time

On Nonconvex Optimization Problems with D.C. Equality and Inequality Constraints

Optimal insurance contract specification in the upstream sector of the oil and gas industry

We consider minimizing a class of nonconvex composite stochastic optimization problems, and deterministic optimization problems whose objective function consists of an expectation function (or an average of finitely many smooth functions) and a weakly convex but potentially nonsmooth function. And in this paper, we focus on the theoretical properties of two types of stochastic splitting methods for solving these nonconvex optimization problems: stochastic alternating direction method of multipliers and stochastic proximal gradient descent. In particular, several inexact versions of these two types of splitting methods are studied. At each iteration, the proposed schemes inexactly solve their subproblems by using relative error criteria instead of exogenous and diminishing error rules, which allows our approaches to handle some complex regularized problems in statistics and machine learning. Under mild conditions, we obtain the convergence of the schemes and their computational complexity related to the evaluations on the component gradient of the smooth function, and find that some conclusions of their exact counterparts can be recovered.

Nonconvex optimization problems, such as those often seen in chemical engineering applications due to integer decisions and inherent system physics, are known to scale poorly with problem size. Decomposition methods, such as Benders or Lagrangean decomposition, offer much promise for improving solution efficiency. However, automatically identifying subproblems for use with these solution methods remains an open problem. Herein, a general algorithmic framework entitled Detection of Communities for Optimization Decomposition (DeCODe) is presented. DeCODe uses community detection, a concept originating from network theory, to generate optimization subproblems which are strongly interacting within individual subproblems but weakly interacting between different ones. The importance of communities in solving problems using decomposition solution methods is showcased via a least squares regression problem. The ability of DeCODe to identify nontrivial decompositions of optimization problems is demonstrated through a large renewable energy and chemical production optimal design problem and two mixed integer nonlinear program test problems.

Network loss minimization with voltage security using differential evolution

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. We hope that an insight into the inner workings of these methods will allow the reader to appreciate the unique marriage of task structure and generative models that allow these heuristic techniques to (provably) succeed. The monograph will lead the reader through several widely used nonconvex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.

Service-oriented virtual network deployment is based on statistical resource demands of different services, while data traffic from each service fluctuates over time. In this paper, a delay-aware flow migration problem for embedded services is studied to meet end-to-end (E2E) delay requirement with time-varying traffic. A non-convex multi-objective mixed integer optimization problem is formulated, addressing the trade-off between maximum load balancing and minimum reconfiguration overhead due to flow migrations, under processing and transmission resource constraints and QoS requirement constraints. Since the original problem is non-solvable in optimization solvers due to unsupported types of quadratic constraints, it is transformed to a tractable mixed integer quadratically constrained programming (MIQCP) problem. The optimality gap between the two problems is proved to be zero, so we can obtain the optimum of the original problem through solving the MIQCP problem with some post-processing. Numerical results are presented to demonstrate the aforementioned trade-off, as well as the benefit from flow migration in terms of E2E delay performance guarantee.

In this paper, we study the multiobjective co-design problem of optimal valve placement and operation in water distribution networks, addressing the minimization of average pressure and pressure variability indices. The presented formulation considers nodal pressures, pipe flows and valve locations as decision variables, where binary variables are used to model the placement of control valves. The resulting optimization problem is a multiobjective mixed integer nonlinear optimization problem. As conflicting objectives, average zone pressure and pressure variability can not be simultaneously optimized. Therefore, we present the concept of Pareto optima sets to investigate the trade-offs between the two conflicting objectives and evaluate the best compromise. We focus on the approximation of the Pareto front, the image of the Pareto optima set through the objective functions, using the weighted sum, normal boundary intersection and normalized normal constraint scalarization techniques. Each of the three methods relies on the solution of a series of single-objective optimization problems, which are mixed integer nonlinear programs (MINLPs) in our case. For the solution of each single-objective optimization problem, we implement a relaxation method that solves a sequence of nonlinear programs (NLPs) whose stationary points converge to a stationary point of the original MINLP. The relaxed NLPs have a sparse structure that come from the sparse water network graph constraints. In solving the large number of relaxed NLPs, sparsity is exploited by tailored techniques to improve the performance of the algorithms further and render the approaches scalable for large scale networks. The features of the proposed scalarization approaches are evaluated using a published benchmarking network model.

In this paper, we focus on a cross-layer resource allocation problem for an opportunistic cognitive radio network, where secondary users (SUs) share a primary network’s licensed spectrum only when the primary network is sensed to be idle. We consider cooperative spectrum sensing, where SUs participate in sensing only when a benefit from resource allocation is guaranteed. Number of SUs for cooperative spectrum sensing is an important parameter as it defines both sensing and resource allocation performance. The fusion centre captures interaction between physical layer spectrum sensing, channel condition, and higher layer data buffer while selecting SUs for cooperative sensing. We consider heterogeneous data types at SUs’ data buffers during SU selection to make our system model more general. We form a mixed integer non-linear optimization problem, from which we select SUs’ set for sensing and resource allocation, appropriate sensing thresholds, and corresponding resource allocation parameters. Due to it’s combinatorial nature, the optimization problem is non-trivial. We devise an optimal algorithm to solve the optimization problem. Moreover, we also propose a computationally efficient and near-optimal greedy algorithm with suitable performance bounds. We also show advantage of our proposed optimal algorithms compared to other traditional methods through extensive simulations.