General Optimality Conditions Research Articles

A relaxation viewpoint to Unbalanced Optimal Transport: Duality, optimality and Monge formulation

We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex envelope of a cost for non-negative Dirac masses. New general primal-dual formulations, optimality conditions, and metric-topological properties are carefully studied and discussed.

Actuarial pricing with financial methods

The objective of this paper is twofold. On the one hand, the optimal combination of reinsurance and financial investment will be studied under a general framework. Indeed, there is no specific type of reinsurance contract, there is no specific dynamics of the involved financial instruments and the financial market does not have to be free of frictions. On the other hand, it will be pointed out how the optimal combination above may provide us with new premium principles making the insurer global risk vanish. The risk will be managed with a coherent risk measure, and the new premium principles will seem to reflect several properties, which are desirable from both the analytical and the economic perspectives. From the analytical viewpoint, the premium principles will be continuous, homogeneous and increasing. From the economic viewpoint, the premium principles will lead to cheaper prices with respect to both the insurance market and the financial one. In other words, the premium principles will make the insurer more competitive in prices under a null risk. General necessary and sufficient optimality conditions will be given, as well as closed forms for the solutions under appropriate assumptions. Several methods preventing unbounded optimization problems will warrant special attention, and one particular case will be more thoroughly studied, namely, the combination of the Black–Scholes–Merton pricing model with the conditional value at risk.

Application of the Tomtit Flock Metaheuristic Optimization Algorithm to the Optimal Discrete Time Deterministic Dynamical Control Problem

A new bio-inspired method for optimizing the objective function on a parallelepiped set of admissible solutions is proposed. It uses a model of the behavior of tomtits during the search for food. This algorithm combines some techniques for finding the extremum of the objective function, such as the memory matrix and the Levy flight from the cuckoo algorithm. The trajectories of tomtits are described by the jump-diffusion processes. The algorithm is applied to the classic and nonseparable optimal control problems for deterministic discrete dynamical systems. This type of control problem can often be solved using the discrete maximum principle or more general necessary optimality conditions, and the Bellman’s equation, but sometimes it is extremely difficult or even impossible. For this reason, there is a need to create new methods to solve these problems. The new metaheuristic algorithm makes it possible to obtain solutions of acceptable quality in an acceptable time. The efficiency and analysis of this method are demonstrated by solving a number of optimal deterministic discrete open-loop control problems: nonlinear nonseparable problems (Luus–Tassone and Li–Haimes) and separable problems for linear control dynamical systems.

Linear Versus Nonlinear (Convex and Concave) Hedging Rules for Reservoir Optimization Operation

AbstractHedging rules of reservoir operations decide the timing and magnitude of current water delivery reduction and consequent carryover water storage conservation based on the tradeoff between the utility of current and future water use. Different forms of hedging can be used for reservoir release decision. This study presents general optimality conditions for hedging with convex, concave, and linear relationships between reservoir water availability and release, using criteria based on the relative value of marginal value of risk tolerance (MVRT) with respect to current reservoir release versus carryover storage. MVRT is quantified to measure reservoir operators' risk tolerance change (sensitivity) to the change of reservoir release or carryover storage, reflecting their attitudes to the risk with current release and that with the carryover storage (i.e., future release), as well as their perception of future water availability uncertainty. Higher, equal, or lower MVRT with current release than that with carryover storage for future release corresponds to convex, linear or concave hedging. Various levels of risk tolerance toward hydrologic uncertainty underlie different hedging types for reservoir release decisions. Specifically, negative, positive, and null risk premiums (or risk seeking, risk aversion, and risk neutrality) result from convex, concave, and linear hedging rules, respectively. In general, high uncertainty will move the convex or concave curve closer to the linear curve. The MVRT‐based criteria for optimal hedging policies are illustrated through a real‐world case study.

Retraction Note: Existence of global solutions to a quasilinear Schr\xf6dinger equation with general nonlinear optimal control conditions

This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1186/s13661-021-01502-z

Actuarial Pricing with Nancial Methods

The objective of this paper is twofold. On the one hand, the optimal combination of reinsurance and financial investment is studied under a general framework, since there is no specific type of reinsurance contract, there is no specific dynamics of the financial instruments, the financial market does not have to be free of frictions and there are no constraints with respect to the financial securities to be selected, that is, derivative assets may be involved too. On the other hand, it is pointed out how the optimal combination above may provide us with new premium principles making the global insurer risk vanish, this risk being computed with a co- coherent risk measure. The new premium principles seem to respect several properties which are desirable from both the analytical and the economic perspectives. Indeed, from an analytical viewpoint, they are continuous, homogeneous, and increasing, while from the economic viewpoint the premium principles lead to cheaper prices with respect to both the insurance market and the financial market. In other words, the premium principles make the insurer more competitive though the insurer's global risk is vanishing. General necessary and sufficient optimality conditions will be given, as well as closed forms for the solutions under appropriate assumptions. Special attention will deserve those methods preventing unbounded optimization problems and a particular case will be more thoroughly studied, namely, the combination of the Black-Scholes- Merton pricing model with the conditional value at risk.

Constrained nonsmooth problems of the calculus of variations

The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary and isoperimetric constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point, when subdifferential-based optimality conditions fail to disqualify this point as non-optimal.

On optimal solutions of general continuous‐singular stochastic control problem of McKean‐Vlasov type

In this paper, we establish general necessary optimality conditions for stochastic continuous‐singular control of McKean‐Vlasov type equations. The coefficients of the state equation depend on the state of the solution process as well as of its probability law and the control variable. The coefficients of the system are nonlinear and depend explicitly on the absolutely continuous component of the control. The control domain under consideration is not assumed to be convex. The proof of our main result is based on the first‐ and second‐order derivatives, with respect to measure in Wasserstein space of probability measures, and by using variational method.

RETRACTED ARTICLE: Existence of global solutions to a quasilinear Schr\xf6dinger equation with general nonlinear optimal control conditions

In this article, we study a modified maximum principle approach under condition on the weight of the delay term in the feedback and the weight of the term without delay. On that basis, we prove the existence of global solutions for a quasilinear Schrödinger equation in an unbounded domain with a general nonlinear nonlinear optimal control condition in the weakly nonlinear internal feedback. The equation includes many special cases such as classical Schrödinger equations, fractional Schrödinger equations, and relativistic Schrödinger equations, etc. Our results are established by means of the fixed point theory associated with the Schrödinger operator in suitable b-metric spaces. Moreover, we establish general stability estimates by using some properties of Schrödinger convex functions.

Modeling, simulation and optimization of combined fractional-ordinary dynamic systems

This work proposes an approach to the modeling and optimization of systems involving a coupled set of fractional and ordinary differential equations. We first present a generalized version of the predictor-corrector integration method, which can integrate simultaneously both fractional and ordinary differential equations. Further, we describe an analytical/numerical dynamic optimization strategy that combines the generalized optimality conditions for a fractional-ordinary system derived in this work, the generalized integration technique and the gradient method. The approach is illustrated through a compartmental model in pharmacokinetics as well as a fractional model for a thermal hydrolysis. In both cases, after we apply a formal fractionalization strategy, we propose a reformulation of the models to obtain fractional-ordinary dynamic systems. The systems obtained are further posed within an optimization framework and solved through our approach as fractional-ordinary optimal control problems. Our results show the theoretical and numerical consistency of our approach.

Regularity result for a shape optimization problem under perimeter constraint

We study the problem of optimizing the eigenvalues of the Dirichlet Laplace operator under perimeter constraint. We prove that optimal sets are analytic outside a closed singular set of dimension at most $d-8$ by writing a general optimality condition in the case the optimal eigenvalue is multiple. As a consequence we find that the optimal $k$-th eigenvalue is strictly smaller than the optimal $(k+1)$-th eigenvalue. We also provide an elliptic regularity result for sets with positive and bounded weak curvature.

Secrecy-Oriented Antenna Assignment Optimization at Full-Duplex Receiver With Self-Interference

This letter investigates a secure MIMO system, in which a full-duplex receiver ( Bob ) assigns part of his antennas to emit jamming signals to confound the passive eavesdropper ( Eve ). Unlike previous antenna assignment schemes based on the idealized assumption that the self-interference (SI) can be canceled perfectly, we consider and quantify the contradicting effects of jamming signals acting as SI signals for Bob and confusing signals for Eve , and then formulate the antenna assignment optimization as an integer programming problem. By balancing the contradicting effects, we derive a closed-form solution and further reveal the general design guidelines, i.e., the feasibility and optimality conditions, for the considered system. Numerical results verify our theoretical analysis and demonstrate the superior performance of the proposed scheme.

Derivation of operation rules for reservoirs in parallel with joint water demand

AbstractThe purpose of this paper is to derive the general optimality conditions of the commonly used operating policies for reservoirs in parallel with joint water demand, which are defined in terms of system‐wide release rules and individual reservoir storage balancing functions. Following that, a new set of release rules for individual reservoirs are proposed in analytical forms by considering the optimality conditions for the balance of total water delivery utility and carryover storage value of individual reservoirs. Theoretical analysis indicates that the commonly used operating policies are a special case of the newly derived rules. The derived release rules are then applied to simulating the operation of a parallel reservoir system in northeastern China. Compared to the performance of the commonly used policies, some advantages of the proposed operation rules are illustrated. Most notably, less water shortage occurrence and higher water supply reliability are obtained from the proposed operation rules.

Channel Coordination of a Risk-Averse Supply Chain: Mean-Variance Approach

We investigate the risk-averse supply chain comprising one supplier and one retailer with mean-variance objective function. We derive a general optimality condition that holds for an integrated supply chain as well as a decentralized one, for channel coordinating contracts like buyback and revenue-sharing. Deviating from the established coordination mechanism for risk-averse supply chain, we demonstrate that a supply chain agent’s mean-variance objective optimization depends only on the demand distribution and is independent of the model parameter. The proposed method also has computational advantage over the existing mechanisms for calculating the optimal order quantity. We further establish the conditions under which one agent bears more risk than the other for aforementioned contracts.

Sufficient optimality conditions in hierarchical models of nonuniform systems

We give a brief survey of the studies developed on the basis of general Krotov's sufficient optimality conditions in an important research direction related to nonuniform systems. We construct a two-layered model of the network structure. The upper level of this model is an abstract network of operators; the lower level contains continuous dynamical models. For such a network, we pose an optimization problem and find general sufficient optimality conditions as generalizations of sufficient conditions for discrete-continuous dynamical systems. We survey possible applications of this direction and consider an example of optimizing nature preservation activities in a river basin.

Mixed Strategies in Discriminatory Divisible-Good Auctions

Using the concept of market-distribution functions, we derive general optimality conditions for discriminatory divisible-good auctions, which are also applicable to Bertrand games and non-linear pricing. We introduce the concept of o¤er distribution function to analyze randomized offer curves, and characterize mixed-strategy Nash equilibria for pay-as-bid auctions where demand is uncertain and costs are common knowledge; a setting for which pure-strategy supply function equilibria typically do not exist. We generalize previous results on mixtures over horizontal offers as in Bertrand-Edgeworth games, and we also characterize novel mixtures over partly increasing supply functions.

Exact algorithms for OWA-optimization in multiobjective spanning tree problems

This paper deals with the multiobjective version of the optimal spanning tree problem. More precisely, we are interested in determining the optimal spanning tree according to an Ordered Weighted Average (OWA) of its objective values. We first show that the problem is weakly NP-hard. We then propose different mixed integer programming formulations, according to different subclasses of OWA functions. Furthermore, we provide various preprocessing procedures, the validity scopes of which depend again on the considered subclass of OWA functions. For designing such procedures, we propose generalized optimality conditions and efficiently computable bounds. These procedures enable to reduce the size of the instances before launching an off-the-shelf software for solving the mixed integer program. Their impact on the resolution time is evaluated on the basis of numerical experiments.

General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity

Motivated by the recent investigations, first a general framework for a class of $(\rho, \eta, A)$-invex n-set functions is introduced, and then some optimality conditions for multiple objective fractional programming on the generalized $(\rho, \eta, A)$-invexity are explored. The obtained results are general in nature and application-oriented to other investigations on fractional subset programming in literature.

Global infimum of strictly convex quadratic functions with bounded perturbations

The problem of minimizing \({\tilde f=f+p}\) over some convex subset of a Euclidean space is investigated, where f(x) = xTAx + bTx is strictly convex and |p| is only assumed to be bounded by some positive number s. It is shown that the function \({\tilde f}\) is strictly outer γ-convex for any γ > γ*, where γ* is determined by s and the smallest eigenvalue of A. As consequence, a γ*-local minimal solution of \({\tilde f}\) is its global minimal solution and the diameter of the set of global minimal solutions of \({\tilde f}\) is less than or equal to γ*. Especially, the distance between the global minimal solution of f and any global minimal solution of \({\tilde f}\) is less than or equal to γ*/2. This property is used to prove a roughly generalized support property of \({\tilde f}\) and some generalized optimality conditions.

A Fast and Effective Routine Method Based on SPME and GC/ICP-MS for the Monitoring of Organotin Compounds in Surface and Sea Water

The quality assessment of water in term of organometallic compounds is one of the key tasks of modern environmental analytical chemistry. For this purpose a rapid and very sensitive method based on solid-phase microextraction (SPME) followed by gas chromatography-inductively coupled plasma-mass spectrometry (GC-ICP-MS) was developed and validated aimed at the determination of dimethyl-, dibutyl- and tributyltin in water samples. A divinylbenzene/ carboxen/polidimethylsiloxane fiber was used for the SPME sample treatment. Coupling of the ICP-MS with GC was accomplished developing an easy-to-fit in-house interface. In general optimal conditions were found by using the enhanced “hot plasma/protective ion extraction” introduction system. Validation was carried out in terms of limit of detection, limit of quantitation, linearity, precision and trueness. GC-ICPMS made possible the detection of the organotin compounds at ultratrace levels, quantitation limits ranging from 0.04 to 5 ng l-1. Linearity was established over two orders of magnitude for all the investigated compounds. Good precision was obtained both in terms of intra-day repeatability and between-day precision on two concentration levels (RSD lower than 15%). Recovery values ranging from 75±2% to 99±1.2% (n = 3) were calculated. The method was proved rapid, GC-ICPMS allowing for the separation of organotin species in less than 7 minutes. The method was used in routine analysis of environmental samples (fresh and marine water samples). Keywords: Dimethyltin, Dibutyltin, Tributyltin, Solid-phase microextraction, Gas chromatography-inductively coupled plasma mass spectrometry, Home-made interface