How to Generate Good Profit Maximization Problems

In this paper, we analyse the performance of the closed-loop Whiplash gradient descent algorithm [1] for L-smooth convex cost functions. Using numerical experiments, we study the algorithm’s performance for convex cost functions, for different condition numbers. We analyse the convergence of the momentum sequence using symplectic integration and introduce the concept of relaxation sequences which analyses the non-classical character of the whiplash method. Under the additional assumption of invexity, we establish a momentum-driven adaptive convergence rate. Furthermore, we introduce an energy method for predicting the convergence rate with convex cost functions for closed-loop inertial gradient dynamics, using an integral anchored energy function and a novel lower bound asymptotic notation, by exploiting the bounded nature of the solutions. Using this, we establish a polynomial convergence rate for the whiplash inertial gradient system, for a family of scalar quadratic cost functions and an exponential rate for a quadratic scalar cost function.

A novel projected two-binary-variable formulation for unit commitment in power systems

In group decision making (GDM), to facilitate an acceptable consensus among the experts from different fields, time and resources are paid for persuading experts to modify their opinions. Thus, consensus costs are important for the GDM process. Notwithstanding, the unit costs in the common linear cost functions are always fixed, yet experts will generally express more resistance if they have to make more compromises. In this study, we use the quadratic cost functions, the marginal costs of which increase with the opinion changes. Aggregation operators are also considered to expand the applications of the consensus methods. Moreover, this paper further analyzes the minimum cost consensus models under the weighted average (WA) operator and the ordered weighted average (OWA) operators, respectively. Corresponding approaches are developed based on strictly convex quadratic programming and some desirable properties are also provided. Finally, some examples and comparative analyses are furnished to illustrate the validity of the proposed models.

Structural sparsity is useful for variable and node selection in distributed networks. In this paper, we propose a distributed algorithm to solve the problem of a quadratic cost function with mixed $\ell_{1,2}$ -norm regularization to promote the group-sparsity of the solution. By introducing virtual pair nodes to each actual node and by decomposing the cost function to each nodes, we obtain a distributed optimization problem on an extended graph model, which is further solved via the PDMM algorithm. Numerical simulation results illustrate the accurate convergence of the proposed algorithm to the centralized solution.

In this paper we are concerned with the problem of sequencing a given set of jobs without preemption on a single machine so as to minimize total cost, where associated with each job is a processing time and a differentiable cost function defined on the completion time of the job. The problem, in general, is NP-complete and, therefore, there is unlikely to be an algorithm to solve the problem in reasonable time, thus a heuristic algorithm is desirable. We present two heuristic algorithms to solve the problem. The first algorithm is based on the differential of the cost functions, and the second algorithm is based on the least square approximation of the cost functions. Computational experiences for the case of quadratic, cubic, and exponential cost functions are presented.

We study the effects of merging two separate markets each originally monopolized by a producer into a globalized duopoly market. We consider a linear inverse demand with cap price and quadratic cost functions. After globalization, we find the consistent conjectural variations equilibrium (CCVE) of the duopoly game. Unlike in the Cournot equilibrium, a complete symmetry (identical cost functions parameters of both firms) does not imply the strongest coincident profit degradation. For the situation where both agents are low-marginal cost firms, we find that the company with a technical advantage over her rival has a better ratio of the current and previous profits. Moreover, as the rival becomes ever weaker, that is, as the slope of the rival’s marginal cost function increases, the profit ratio improves.

In this article, a linear quadratic tracking controller is designed for continuous‐time delay systems involving multiplicative noises. This problem is challenging since the separation principle is ineffective for Itô stochastic systems and it is not assumed that the control weighting matrix in the cost function is positive definite. In this work, both finite time horizon and infinite time horizon cases are considered. The optimal controller is designed such that quadratic cost function in terms of the tracking error of the states and the control input is minimized. It is noted that comparing with regular problem, that is, the weighting matrix of control input in cost function is positive semidefinite, there exists an extra term in the optimal controller which can be free to choose. The explicit analytical expressions for the optimal controller are provided through solving a couple of Riccati‐like equations. For the infinite horizon case, the stabilization problem for an Itô stochastic system containing the input time delay is also studied. The effectiveness of the obtained theoretical results is illustrated by a numerical example.

The shape of cost functions significantly influences the solution of optimization problems related to the treatment of infectious diseases. The classical linear or quadratic cost functions are widely used in the literature more to make the problems tractable than for their relevance with the model studied. For example, they do not seem appropriate when expensive therapies are required to treat a large number of patients, as in the case of the treatment of hepatitis C by sofosbuvir. In this paper, we study the strategic interactions between pharmaceutical companies and public authorities to determine the price of patented drugs and the number of treated infected people. A Stackelberg game and a Nash bargaining game are studied. In particular, we analyze the relationship between the different cost functions and the equilibria of the games.

The paper has set forth and developed a stochastic cost-volume-value (CVV) model with economic demand and cost functions. The model describes the relationship among cost, volume and value under uncertainty; and helps company managers conduct breakeven analysis in management decision-making and planning. Compared with the cost-volume-profit (CVP) model, the function of CVV model is EVA but not profits. Therefore, the model offset the limitation of the conventional CVP model which does not take the capital costs into consideration, and is more suitable for managers to make decisions in value creation. In reality, product prices and unit costs are related with volume. The model herein incorporates a linear demand function and a quadratic cost function. According to the stochastic CVV model, l.Explicit solution is found for the quantity which maximizes expected EVA. 2. The probability expression of expected EVA is found. 3. A specific expression which can be used for determining fixed cost is found, and then managers can choose activities which really create value for enterprises.

In this paper, we develop an online optimization algorithm with integral action for solving online optimization problems characterized by quadratic cost functions with a linearly varying optimal point. The proposed algorithm can be formulated as an uncertain linear feedback system incorporating a discrete time double integrator. By leveraging established results on the optimal gain margin, we demonstrate that the proposed algorithm achieves the optimal convergence rate compared to other methods including a discrete time double integrator applied to quadratic cost functions in finite-dimensional spaces.

Because of employing elevation adaptivity, three-dimensional space-time adaptive processing (3D STAP) can achieve better performance than conventional two-dimensional (2D) STAP at the cost of higher computational load and sample support requirement. To overcome these shortcomings, a tri-iterative 3D STAP method based on correlation matrix for airborne radar is developed. In the proposed method, the quadratic cost function used in the optimum STAP is converted into three quadratic functions by using submatrices of the space-time correlation matrix, and the full dimension weight vector can be separated into three lower dimensional weight vectors. By iteratively optimizing these lower dimensional weight vectors, the proposed method can significantly decrease the computational load and training samples requirement. Experiment results using both simulated data and measured radar data demonstrate the effectiveness of the proposed method.

This paper presents a direct digital control law synthesis procedure for a large order, sampled data, linear feedback system using constrained optimization techniques to meet multiple design requirements. A linear quadratic Gaussian type cost function is minimized while satisfying a set of constraints on the design loads and responses. General expressions for gradients of the cost function and constraints, with respect to the digital control law design variables are derived analytically and computed by solving a set of discrete Liapunov equations. The designer can choose the structure of the control law and the design variables, hence a stable classical control law as well as an estimator-based full or reduced order control law can be used as an initial starting point. Selected design responses can be treated as constraints instead of lumping them into the cost function. This feature can be used to modify a control law, to meet individual root mean square response limitations as well as minimum single value restrictions. Low order, robust digital control laws were synthesized for gust load alleviation of a flexible remotely piloted drone aircraft.

In demand side management electricity consumers are systematically encouraged by a load serving entity (LSE) to adjust their electricity demand to reduce overall costs. Li et al. [1] have presented such an iterative demand response program. In each iteration it sends price signals to the consumers against their demands for each hour of the day ahead, and smart consumer appliances adjust their usage accordingly to maximize their utility given the prices. The price signal is the marginal cost based on a single quadratic and piecewise smooth cost function. We extend previous work to allow multiple cost functions one for each supplier, and require that the LSE optimizes procurement jointly with the demand response. The price signal is then based on the optimal cost of procurement from all suppliers, computed using sequential quadratic programming (SQP). However, when the number of cost functions grows computing the price signal in each iteration becomes a bottleneck. Therefore, we also introduce an approximation technique based on knowledge compilation, in which the procurement problem is first represented in discrete form in propositional logic, and compiled into a tractable form known as decomposable negation normal form (DNNF). The DNNF is then used for efficient computation of near optimal prices in each iteration.

This paper presents a Bayesian approach for the optimal control of a machine that can experience setup errors assuming an asymmetric off-target cost function. It is assumed that the setup error cannot be observed directly due to presence of measurement and part-to-part errors, and it is further assumed that the variance of this error is not known a priori. The setup error can be compensated by performing sequential adjustments of the process mean based on observations of the parts produced. It is shown how the proposed method converges to the optimal (known variance) trajectory, recovering from a possibly biased initial variance estimate. Simulations results are presented to show the small sample behavior of the proposed method under two types of asymmetric off-target cost functions: a constant asymmetric and a quadratic asymmetric cost function.

We consider the problem to control a large population of noncooperative heterogeneous agents, each with strongly convex quadratic cost function depending on the average population state, and all sharing a convex constraint, towards a competitive equilibrium. We assume a minimal information structure through which a central controller can broadcast incentive signals to control the decentralized optimal responses of the agents. We propose a model-free dynamic control law that, based on monotone operator theory arguments, ensures global convergence to an equilibrium independently on the parameters defining the quadratic cost functions, nor on the convex constraints.