New technique for solving univariate global optimization

summary:In this paper, a new global optimization method is proposed for an optimization problem with twice differentiable objective function a single variable with box constraint. The method employs a difference of linear interpolant of the objective and a concave function, where the former is a continuous piecewise convex quadratic function underestimator. The main objectives of this research are to determine the value of the lower bound that does not need an iterative local optimizer. The proposed method is proven to have a finite convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering methods

Outer Approximation Algorithms for DC Programs and Beyond

We consider the well-known Canonical DC (CDC) optimization problem, relying on an alternative equivalent formulation based on a polar characterization of the constraint, and a novel generalization of this problem, which we name Single Reverse Polar problem (SRP). We study the theoretical properties of the new class of (SRP) problems, and contrast them with those of (CDC)problems. We introduce of the concept of ``approximate oracle'' for the optimality conditions of (CDC) and (SRP), and make a thorough study of the impact of approximations in the optimality conditions onto the quality of the approximate optimal solutions, that is the feasible solutions which satisfy them. Afterwards, we develop very general hierarchies of convergence conditions, similar but not identical for (CDC) and (SRP), starting from very abstract ones and moving towards more readily implementable ones. Six and three different sets of conditions are proposed for (CDC) and (SRP), respectively. As a result, we propose very general algorithmic schemes, based on approximate oracles and the developed hierarchies, giving rise to many different implementable algorithms, which can be proven to generate an approximate optimal value in a finite number of steps, where the error can be managed and controlled. Among them, six different implementable algorithms for (CDC) problems, four of which are new and can't be reduced to the original cutting plane algorithm for (CDC) and its modifications; the connections of our results with the existing algorithms in the literature are outlined. Also, three cutting plane algorithms for solving (SRP) problems are proposed, which seem to be new and cannot be reduced to each other

Productivity enhancement through process integration

A hierarchical procedure is developed to determine maximum overall yield of a process and optimize process changes to achieve such a yield. First, a targeting procedure is developed to identify an upper bound of the overall yield ahead of detailed design. Several mass integration strategies are proposed to attain maximum yield. These strategies include rerouting of raw materials, optimization of reaction yield, rerouting of product from undesirable outlets to desirable outlets, and recycling of unreacted raw materials. Path equations are tailored to provide the appropriate level of detail for modeling process performance as a function of the optimization variables pertaining to design and operating variables. Interval analysis is used as an inclusion technique that provides rigorous bounds regardless of the process nonlinearities and without enumeration. Then, a new approach for identification of cost-effective implementation of maximum attainable targets for yield is presented. In this approach, a mathematical program was developed to identify the maximum feasible yield using a combination of iterative additions of constraints and problem reformulation. Next, cost objectives were employed to identify a cost-effective solution with the details of design and operating variables. Constraint convexification was used to improve the quality of the solution towards globability. A trade-off procedure between the saving and expenses for yield maximization problem is presented. The proposed procedure is systematic, rigorous, and computationally efficient. A case study was solved to demonstrate the applicability and usefulness of the developed procedure

Nonparametric Option Pricing under Shape Restrictions

Frequently, economic theory places shape restrictions on functional relationships between economic variables. This paper develops a method to constrain the values of the first and second derivatives of nonparametric locally polynomial estimators. We apply this technique to estimate the state price density (SPD), or risk-neutral density, implicit in the market prices of options. The option pricing function must be monotonic and convex. Simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day-to-day option pricing, once appropriate theory-motivated shape restrictions are imposed. Using S&P500 option prices, we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999, unlike the constrained estimator we propose.

The link approach to measuring consumer surplus in transport networks

Should one calculate user benefits from changes in door-to-door journeys or from changes in the use of separate links of the network? The second approach is often discarded for its perceived inability to deal with new links and the OD-matrix approach is favoured. Differences arise when the set of used routes changes. A consumer model containing a general static transportation network with explicit non-negativity constraints serves as a basis for welfare measures expressed in shadow prices. The approximation error when applying the link approach need not be too severe. A rehabilitation of the link approach may be in order.

Optimal Resource Allocation for Multi-user OFDMA-URLLC MEC Systems

In this paper, we study resource allocation algorithm design for multi-user orthogonal frequency division multiple access (OFDMA) ultra-reliable low latency communication (URLLC) in mobile edge computing (MEC) systems. To meet the stringent end-to-end delay and reliability requirements of URLLC MEC systems, we propose joint uplink-downlink resource allocation and finite blocklength transmission. Furthermore, we employ a partial time overlap between the uplink and downlink frames to minimize the end-to-end delay, which introduces a new time causality constraint. The proposed resource allocation algorithm is formulated as an optimization problem for minimization of the total weighted power consumption of the network under a constraint on the number of URLLC user bits computed within the maximum allowable computation time, i.e., the end-to-end delay of a computation task. Despite the non-convexity of the formulated optimization problem, we develop a globally optimal solution using a branch-and-bound approach based on discrete monotonic optimization theory. The branch-and-bound algorithm minimizes an upper bound on the total power consumption until convergence to the globally optimal value. Furthermore, to strike a balance between computational complexity and performance, we propose two efficient suboptimal algorithms based on successive convex approximation and second-order cone techniques. Our simulation results reveal that the proposed resource allocation algorithm design facilitates URLLC in MEC systems, and yields significant power savings compared to three baseline schemes. Moreover, our simulation results show that the proposed suboptimal algorithms offer different trade-offs between performance and complexity and attain a close-to-optimal performance at comparatively low complexity.Comment: 32 pages, 9 figures, submitted for an IEEE journal. arXiv admin note: substantial text overlap with arXiv:2005.0470