Lower-bounding Problem Research Articles

Computing subgradients of convex relaxations for solutions of parametric ordinary differential equations

A novel subgradient evaluation method is proposed for nonsmooth convex relaxations of parametric solutions of ordinary differential equations (ODEs) arising in global dynamic optimization, assuming that the relaxations always lie strictly within interval bounds during integration. We argue that this assumption is reasonable in practice. These subgradients are computed as the unique solution of an auxiliary parametric affine ODE, analogous to classical forward/tangent sensitivity evaluation methods for smooth dynamic systems. Unlike established subgradient evaluation approaches for nonsmooth dynamic systems, this new method does not require smoothness or transversality assumptions, and is compatible with existing subgradient evaluation methods for closed-form convex functions, as implemented in subgradient evaluation software such as EAGO.jl and MC ++. Moreover, we show that a subgradient for a lower-bounding problem in global dynamic optimization can be directly evaluated using reverse/adjoint sensitivity analysis, which may reduce the overall computational effort for an overarching global optimization method. Numerical examples are presented, based on a proof-of-concept implementation in Julia.

Axisymmetric Contact Problems for Composite Pressure Vessels

The present analysis is conducted for the evaluation of contact pressures of axisymmetric shells made of laminated composites or functionally graded materials. This class of problems is usually called the Signorini–Fichera problem (unilateral constraints) and can be solved as the lower-bound problem. The numerical solution of this problem is proposed both for symmetric and unsymmetric shell configurations. The first-ply-failure of such structures is considered. It is demonstrated that the failure occurs at the end of the contact area corresponding to the appearance of stress concentration of radial concentrated forces.

Deterministic global superstructure-based optimization of an organic Rankine cycle

Organic Rankine cycles (ORCs) offer a high structural design flexibility. The best process structure can be identified via the optimization of a superstructure, which considers design alternatives simultaneously. In this contribution, we apply deterministic global optimization to a geothermal ORC superstructure, thus guaranteeing to find the best solution. We implement a hybrid mechanistic data-driven model, employing artificial neural networks as thermodynamic surrogate models. This approach is beneficial as we optimize the problem in a reduced space using the optimization solver MAiNGO. We further introduce redundant constraints that are only considered for the lower-bounding problem of the branch-and-bound algorithm. We perform two separate optimizations, one maximizing power output and one minimizing levelized cost of electricity. The optimal solutions of both objectives differ from each other, but both have three pressure levels. Global optimization is necessary as there exist suboptimal local solutions for both flowsheet configuration and design with fixed configurations.

Resource Allocation and Basestation Placement in Downlink Cellular Networks Assisted by Multiple Wireless Powered UAVs

In this paper, we focus on a downlink cellular network, where multiple UAVs serve as aerial basestations to provide wireless connectivity to ground users through frequency division multi-access (FDMA) scheme. The UAVs are exclusively powered by a wireless charging station located on the ground following save-then-transmit protocol. In such a UAV-assisted cellular network, joint optimization for user association, resource allocation and basesation placement are investigated to maximize the downlink sum rate. The problem is formulated as a mixed integer optimization problem and is thus challenging to solve. We propose an efficient solution based on alternating optimization, by iteratively solving one of the three subproblems (i.e., user association, resource allocation and basesation placement) with the other two fixed. Specificly, user association is solved as a standard linear programming problem by relaxing the binary association indicators into continuous variables. For basestation placement and resource allocation, we resort to successive convex optimization technique, which iteratively solves a lower-bound problem. After iteratively solving the three subproblems, we further propose an algorithm based on penalty method and successive convex optimization to make the association indicators feasibly binary. We conduct comprehensive experiments for the optimal solution to the three subproblems with insightful results. We also show that the optimal downlink sum rate cannot be always enhanced by deploying more UAVs due to non-negligible tradeoff between energy/communication sources and co-channel interference. Moreover, the proposed solution outperforms a baseline strategy leveraged from an existing work, especially with favorable channel condition and sufficient frequency resources.

Eigen Value Analysis in Lower Bounding Uncertainty of Kalman Filter Estimates

In this paper we are concerned with the error-covariance lower-bounding problem in Kalman filtering: a sensor releases a set of measurements to the data fusion/estimation center, which has a perfect knowledge of the dynamic model, to allow it to estimate the states, while preventing it to estimate the states beyond a given accuracy. We propose a measurement noise manipulation scheme to ensure lower-bound on the estimation accuracy of states. Our proposed method ensures lower-bound on the steady state estimation error of Kalman filter, using mathematical tools from eigen value analysis.

Effects of unconventional monetary policy on income and wealth distribution: Evidence from United States and Eurozone

As an answer to the ?Great Recession? and Zero Lower Bound problem, main central banks had to use unconventional monetary policy (UMP). This research focuses on the distributive effects of these measures on household income and household wealth in the United States of America (USA) and the Eurozone. For this purpose, this paper presents four models that were constructed using the Structural Vector Autoregressive methodology (SVAR). The results suggest that the UMPs applied by the Federal Reserve (FED) in the USA could increase wealth and income inequality through the portfolio channel. However, the same results were not observed in the Eurozone.

Global Solution Strategies for the Network-Constrained Unit Commitment Problem With AC Transmission Constraints

We propose a novel global solution algorithm for the network-constrained unit commitment problem incorporating a nonlinear alternating current model of the transmission network, which is a nonconvex mixed-integer nonlinear programming (MINLP) problem. Our algorithm is based on the multi-tree global optimization methodology, which iterates between a mixed-integer lower-bounding problem and a nonlinear upper-bounding problem. We exploit the mathematical structure of the unit commitment problem with AC power flow constraints (UC-AC) and leverage optimization-based bounds tightening, second-order cone relaxations, and piecewise outer approximations to guarantee a globally optimal solution at convergence. Numerical results on four benchmark problems illustrate the effectiveness of our algorithm, both in terms of convergence rate and solution quality.

A multitree approach for global solution of ACOPF problems using piecewise outer approximations

Electricity markets rely on the rapid solution of the optimal power flow (OPF) problem to determine generator power levels and set nodal prices. Traditionally, the OPF problem has been formulated using linearized, approximate models, ignoring nonlinear alternating current (AC) physics. These approaches do not guarantee global optimality or even feasibility in the real ACOPF problem.We introduce an outer-approximation approach to solve the ACOPF problem to global optimality based on alternating solution of upper- and lower-bounding problems. The lower-bounding problem is a piecewise relaxation based on strong second-order cone relaxations of the ACOPF, and these piecewise relaxations are selectively refined at each major iteration through increased variable domain partitioning. Our approach is able to efficiently solve all but one of the test cases considered to an optimality gap below 0.1%. Furthermore, this approach opens the door for global solution of MINLP problems with AC power flow equations.

Learning in Combinatorial Optimization: What and How to Explore

We study dynamic decision-making under uncertainty when, at each period, a decision-maker implements a solution to a combinatorial optimization problem. The objective coefficient vectors of said problem, which are unobserved prior to implementation, vary from period to period. These vectors, however, are known to be random draws from an initially unknown distribution with known range. By implementing different solutions, the decision-maker extracts information about the underlying distribution, but at the same time experiences the cost associated with said solutions. We show that resolving the implied exploration versus exploitation trade-off efficiently is related to solving a Lower Bound Problem (LBP), which simultaneously answers the questions of what to explore and how to do so. We establish a fundamental limit on the asymptotic performance of any admissible policy that is proportional to the optimal objective value of the LBP problem. We show that such a lower bound might be asymptotically attained by policies that adaptively reconstruct and solve LBP at an exponentially decreasing frequency. Because LBP is likely intractable in practice, we propose policies that instead reconstruct and solve a proxy for LBP, which we call the Optimality Cover Problem (OCP). We provide strong evidence of the practical tractability of OCP which implies that the proposed policies can be implemented in real-time. We test the performance of the proposed policies through extensive numerical experiments and show that they significantly outperform relevant benchmarks in the long-term and are competitive in the short-term.

Learning in Combinatorial Optimization: What and How to Explore

When moving from the traditional to combinatorial multiarmed bandit setting, addressing the classical exploration versus exploitation trade-off is a challenging task. In “Learning in Combinatorial Optimization: What and How to Explore,” Modaresi, Sauré, and Vielma show that the combinatorial setting has salient features that distinguish it from the traditional bandit. In particular, combinatorial structure induces correlation between cost of different solutions, thus raising the questions of what parameters to estimate and how to collect and combine information. The authors answer such questions by developing a novel optimization problem called the lower-bound problem (LBP). They establish a fundamental limit on asymptotic performance of any admissible policy and propose near-optimal LBP-based policies. Because LBP is likely intractable in practice, they propose policies that instead solve a proxy for LBP, which they call the optimality cover problem (OCP). They provide strong evidence of practical tractability of OCP and illustrate the markedly superior performance of OCP-based policies numerically.

The Cointegration Analysis of the Long-term Bond Rates in Japan under the Zero Lower Bound Problems

In April of 2013, Bank of Japan introduced Qualitative and Quantitative Easing (QQE). And the unexpected announcement of additional QQE in October of 2014 surprised many market participants. Just after both announcements stock prices of Japanese market rose rapidly. BOJ had introduced non-traditional monetary policies not only QQE, but Zero Interest Rate Policy (ZIRP) and Quantitative Easing (QE) for about 15 years. Not only Japan but U.S. Federal Reserve, Bank of England of U.K. (BOE) and Sveriges Riksbank of Sweden have also introduced QE. Especially, Federal Reserve Board of U.S. had executed the non-traditional monetary policies of QE1, QE2 and QE3 since November of 2008 till October of 2014 by the chairmen, Bernanke and Yellen. It continues another non-traditional monetary policy, ZIRP under the slowly recovering economy. Meanwhile ECB announced their introduction of QE on January 22 of 2015 and began buying government bonds on March 9, when Europe’s fragile recovery is lagging the rest of the world and a drop in prices is threatening to make things worse. However none of them had not brought adequate result by the non-traditional monetary policies yet. In this thesis we analyze the relationship between Japanese long-term interest rates and its non-traditional monetary policies under persistent and serious deflation. We investigate the extent to which change rates of monetary base and amounts of loan outstanding of banks and other proxy variables affect both the nominal and real interest rates of 10 year JGBs. We used a fundamental bond rate model based on Sargent (1969) and test the cointegrating relationship among the variables. Then we estimate the model with cointegrating relationship by Dynamic OLS and Error Correction model by GMM.

Production Planning with Shortfall Hedging, under Partial Information and Budget Constraint

We study production planning integrated with risk hedging by considering shortfall as the risk measure. In addition to the one-time production quantity decision, there is a real-time hedging strategy throughout the horizon; and the goal is to minimize the gap between a pre-specified target and the total terminal wealth achieved by both production and hedging. We assume partial information --- hedging is executed based on information from the financial market only, and impose a budget constraint to cap any loss from hedging. To find the optimal hedging strategy, we construct a dual problem, which provides a lower bound to the original problem. Solving the lower-bound problem yields the optimal terminal wealth from hedging; the real-time hedging strategy is then mapped out via martingale representation theorem. Interestingly, the optimal hedging strategy takes the form of a portfolio of two options, a digital option and a put option. With the hedging strategy optimized, we show that optimizing production quantity is a convex minimization problem. With both production and hedging optimized, we provide a complete characterization of the efficient frontier: the minimized shortfall as an increasing function of the target. We also derive an explicit quantification of the shortfall reduction achieved by hedging. Asymptotic analysis on several key parameters (such as the target, the budget, and the production quantity) generates additional insights to the hedging strategy and its impact.

Global optimization of nonlinear least-squares problems by branch-and-bound and optimality constraints

We study a simple, yet unconventional approach to the global optimization of unconstrained nonlinear least-squares problems. Non-convexity of the sum of least-squares objective in parameter estimation problems may often lead to the presence of multiple local minima. Here, we focus on the spatial branch-and-bound algorithm for global optimization and experiment with one of its implementations, BARON (Sahinidis in J. Glob. Optim. 8(2):201–205, 1996), to solve parameter estimation problems. Through the explicit use of first-order optimality conditions, we are able to significantly expedite convergence to global optimality by strengthening the relaxation of the lower-bounding problem that forms a crucial part of the spatial branch-and-bound technique. We analyze the results obtained from 69 test cases taken from the statistics literature and discuss the successes and limitations of the proposed idea. In addition, we discuss software implementation for the automation of our strategy.

Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties

In [K. D. Mulmuley and M. Sohoni, SIAM J. Comput., 31 (2001), pp. 496–526], henceforth referred to as Part I, we suggested an approach to the P vs. $NP$ and related lower bound problems in complexity theory through geometric invariant theory. In particular, it reduces the arithmetic (characteristic zero) version of the $NP \not \subseteq P$ conjecture to the problem of showing that a variety associated with the complexity class $NP$ cannot be embedded in a variety associated with the complexity class P. We shall call these class varieties associated with the complexity classes P and $NP$. This paper develops this approach further, reducing these lower bound problems—which are all nonexistence problems—to some existence problems: specifically to proving the existence of obstructions to such embeddings among class varieties. It gives two results towards explicit construction of such obstructions. The first result is a generalization of the Borel–Weil theorem to a class of orbit closures, which include class varieties. The second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class $NC$. These results indicate that the fundamental lower bound problems in complexity theory are, in turn, intimately linked with explicit construction problems in algebraic geometry and representation theory. The results here were announced in [K. D. Mulmuley and M. Sohoni, in Advances in Algebra and Geometry (Hyderabad, $2001$), Hindustan Book Agency, New Delhi, India, 2003, pp. 239–261].

Lower-Bound Solution Algorithm for Equilibrium Signal-Setting Problem

The equilibrium signal-setting problem is stated and subsequently formulated as a continuous equilibrium network design problem. The bilevel formulation is nonconvex and therefore cannot be solved for global optima by using descent solution algorithms. Therefore, a lower bound using a system optimal flow pattern is proposed that will be quite tight in both uncongested and highly congested network traffic situations. A solution algorithm based on the standard steepest-descent method is proposed for the lower-bound problem. Performance of the solution algorithm on a network problem is reported.

Global solution of semi-infinite programs

Optimization problems involving a finite number of decision variables and an infinite number of constraints are referred to as semi-infinite programs (SIPs). Existing numerical methods for solving nonlinear SIPs make strong assumptions on the properties of the feasible set, e.g., convexity and/or regularity, or solve a discretized approximation which only guarantees a lower bound to the true solution value of the SIP. Here, a general, deterministic algorithm for solving semi-infinite programs to guaranteed global optimality is presented. A branch-and-bound (BB the lower-bounding problem is formulated as a convex relaxation of a discretized approximation to the SIP. The SIP B&B algorithm is shown to converge finitely to ɛ−optimality when the subdivision and discretization procedures used to formulate the node subproblems are known to retain certain convergence characteristics. Other than the properties assumed by globally-convergent B&B methods (for finitely-constrained NLPs), this SIP algorithm makes only one additional assumption: For every minimizer x* of the SIP there exists a sequence of Slater points xn for which ** (cf. Section 5.4). Numerical results for test problems in the SIP literature are presented. The exclusion test and a modified upper-bounding problem are also investigated as heuristic approaches for alleviating the computational cost of solving a nonlinear SIP to guaranteed global optimality.

Global Optimization of Nonconvex MINLP by a Hybrid Branch-and-Bound and Revised General Benders Decomposition Approach

Mixed-integer nonlinear programming, MINLP, has played a crucial role in chemical process design via superstructures that always involve discrete and continuous variables. In this paper, a global optimization algorithm for nonconvex MINLP problems is developed by addressing the nonconvexity caused by the nonconvex continuous functions with a convex quadratic underestimator within a branch-and-bound framework, as well as the joint problem caused by the mixed natures of integer and continuous variables through a revised general Benders decomposition (GBD) method, where the latter is designed mainly for three favorable structures, i.e., separable, bilinear, and partly linear, between the two domains of continuous and binary variables. The convergence of the revised GBD method to the global solution of the relaxed MINLP subprobelm over each subregion generated in the above framework is guaranteed by the convex underestimation functions in terms of the twice-differentiable assumptions of the continuous functions and the above three favorable joint structures. Then, the convergence of the proposed hybrid algorithm can be established by the exhaustive partition of the constrained region, the monotonicity of the lower bound, and the reliability of the infeasibility detection. Finally, a very simple example for process design is used to verify the different implementation aspects of the proposed approach, especially the unique underestimator construction and the infeasibility detection in each lower-bounding problem.

Efficient Solution of the Single-item, Capacitated Lot-sizing Problem with Start-up and Reservation Costs

A capacitated dynamic lot-sizing model, where the costs incurred are a start-up cost for switching the production facility on and another reservation cost for keeping the facility on, whether or not it is producing, is considered. The resulting scheduling problem is NP-hard. An efficient shortest path model of the uncapacitated version of the problem is developed. This model is then included, via a redefinition of variables, into a tight capacitated model; tight in the sense that sharp lower bounds can be produced from it. The lower bound problems are solved efficiently by recovering the shortest path structure through column generation, and effective upper bounds are generated by solving a small capacitated trans-shipment problem. The results of computational tests to verify the computational efficiency of the resulting solution scheme are presented.

Some hierarchies for the communication complexity measures of cooperating grammar systems

We investigate here the descriptional and the computational complexity of parallel communicating grammar systems (PCGS). A new descriptional complexity measure — the communication structure of the PCGS is introduced and related to the communication complexity (the number of communications). Several hierarchies resulting from these complexity measures and some relations between the measures are established. The results are obtained due to the development of two lower-bound proof techniques for PCGS. The first one is a generalization of pumping lemmas from formal language theory and the second one reduces the lower-bound problem for some PCGS to the proof of lower bounds on the number of reversals of certain sequential computing models.