On Ratio Invexity in Mathematical Programming

Parametric Solution Algorithms for Large-Scale Mixed-Integer Fractional Programming Problems and Applications in Process Systems Engineering

Energy efficiency (EE) is among the main considerations in the design of modern wireless networks. In this paper, we investigate the EE enhancement for asymmetric analog network coding (ANC) protocol of a two-way relay system based on statistical channel information. A power allocation problem is formulated as the system EE maximization problem with objective function quantified by Goodbit-per-Energy (GPE). Importantly, the EE optimization problem may not be convex and can be categorized into a nonlinear fractional programming problem. Therefore, to solve the problem, a nonlinear fractional programming based algorithm is proposed and closed-form solution is obtained, providing valuable insights into practical system designs. Simulation results highlight the effect of the proposed power allocation.

For a linear fractional programming problem, Sharma and Swarup have constructed a dual problem, also a linear fractional program, in which the objective functions of both primal and dual problems are the same. Craven and Mond have extended this result to a nonlinear fractional programming problem with linear constraints, and a dual problem for which the objective function is the same as that of the primal. This theorem is now further extended from linear to differentiable convex constraints.

Globally convergent exact and inexact parametric algorithms for solving large-scale mixed-integer fractional programs and applications in process systems engineering

In this paper we describe a fuzzy approach for solving nonlinear fractional programming problem with linearconstraints (NLFPP). In the proposed approach, the objective function is transformed into linear function by usingTaylor’s theorem and the considered NLFPP is changed into equivalent linear programming problem (LPP) which itcan be solved as a linear programming problem. The proposed approach is based on choosing three initial points insidethe feasible region which enable us to generate a new point at which the value of objective function is better than theprevious value, and so on to reach the best approximation of optimal solution.

This article addresses the sustainable design and synthesis of open‐loop recycling process of waste high‐density polyethylene (HDPE) under both environmental and economic criteria. We develop by far the most comprehensive superstructure for producing monomers, aromatic mixtures, and fuels from waste HDPE. The superstructure optimization problem is then formulated as a multi‐objective mixed‐integer nonlinear fractional programming (MINFP) problem to simultaneously optimize the unit net present value (NPV) and unit life cycle environmental impacts. A tailored global optimization algorithm integrating the inexact parametric algorithm with the branch‐and‐refine algorithm is applied to efficiently solve the resulting nonconvex MINFP problem. Results show that the optimal unit NPV ranges from $107.2 to $151.3 per ton of HDPE treated. Moreover, the unit life cycle greenhouse gas emissions of the most environmentally friendly HDPE recycling process are 0.40 ton CO2‐eq per ton of HDPE treated, which is 63% of that of the most economically competitive process design.

The sufficient optimality conditions and duality results have recently been given for the following generalised convex programming problem:where the funtion f and g satisfyfor some η: X0 × X0 → ℝnIt is shown here that a relaxation defining the above generalised convexity leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar case, and avoids the major difficulty of verifying that the inequality holds for the same function η(. , .). Further, this relaxation allows one to treat certain nonlinear multi-objective fractional programming problems and some other classes of nonlinear (composite) problems as special cases.

Energy harvesting (EH) is a promising technique for cognitive relaying transmission (CRT) where secondary users (SUs) and relaying nodes do not have a constant power supply each. Unlike conventional CRT where the end-to-end data rate is usually maximised without taking into account the energy consumption at the source and relay, in this study, the energy consumption is characterised by means of energy efficiency, defined as the achievable data rate per Joule. In particular, the energy states at each node (either at a SU or a relay) is modelled as a finite-state Markov chain and the transmit power at a node is optimally allocated by jointly accounting for the interference threshold prescribed by primary users (PUs), the maximum allowable transmit power and the harvested energy at the node. To maximise the energy efficiency, a best relay selection criterion is proposed and the subsequent optimal transmit power allocation is initially formulated as a non-linear fractional programming problem and, then, equivalently transformed into a parametric programming problem and, finally, solved analytically by using the classic Karush–Kuhn–Tucker conditions. With extensive Monte-Carlo simulation results, the effectiveness of the proposed relay selection algorithm and corresponding optimal power allocation strategy are corroborated.

A novel energy efficient opportunistic spectrum access strategy was proposed to solve the energy constrain issue of a secondary user in non-slots cognitive radio network, in which a secondary user sensed the channels licensed to some primary users sequentially before it decided to transmit. To maximize energy efficiency the algorithm can make the secondary user specify the optimal channel sensing time and transmission power to be used upon transmission. During power control process, packet length to be transmitted of the secondary use was considered to regulate the transmission power adaptively. Nonlinear fractional programming problem can be converted into a linear programming problem, thus obtaining the maximum energy efficiency by Binary Search algorithm. Simulation results indicate that the proposed strategy can achieve the tradeoff between the sensing performance and sensing overhead and efficiently improve the energy efficiency of the secondary user.

In conventional digital communication systems, the quality of the received signal does not improve beyond a certain level as the channel quality increases. Such kind of quality saturation effect is caused by the unrecoverable quantization errors produced by source coding. The Hybrid Digital-Analog (HDA) transmission, where the quantization errors are transmitted in an analog mode along with the quantized data in a digital mode, has been recognized as an effective technique to combat the quality saturation effect. In this paper, we introduce HDA transmission in Wireless Relay Networks (WRNs) over Rayleigh slow-fading channels to eliminate the quality saturation effect and achieve graceful improvement for the better channel quality. Our goal is to minimize the end-to-end distortion by optimal power allocation. We note that digital-analog power allocation involved in HDA transmission is coupled with source-relay power allocation in WRNs. Therefore, the joint power allocation problem should be considered. We investigate this problem for two kinds of relays in WRNs, i.e., Amplify-and-Forward (AF) relays and Decode-and-Forward (DF) relays. In the case of AF relays, we find that the joint power problem is concave and thus derive the explicit expressions of the optimal solution. In the case of DF relays, we formulate the joint power allocation problem as a nonlinear fractional programming problem and then propose an efficient algorithm to search the optimal solution. Simulation results show that the proposed joint power allocation schemes outperform existing schemes in terms of end-to-end distortion in both WRNs with AF relays and that with DF relays under various channel conditions.

Multiple objective fractional programming involving semilocally type I-preinvex and related functions

Sustainable Process Design and Synthesis for HDPE Recycling

In this paper, we present a multi-objective fractional probabilistic programming problem where the parameters in the objective functions and right-hand side constraints are multi-choice in nature. Multi-choice multi-objective objective fractional probabilistic programming problem is formulated. The multi-choice parameters in the right-hand side of constraints are random variables following Cauchy distribution. Multi-choice multi-objective fractional probabilistic programming problem is transformed to deterministic equivalent multi-choice multi-objective fractional programming problem using chance-constrained programming problem. Lagrange interpolating polynomial approach is proposed to deal with multi-choice parameters. The multi-choice multi-objective fractional programming problem is transformed to non-linear mixed integer multi-objective programming problem. An \(\epsilon -\) constrained method is applied to find the compromise solutions of non-linear mixed integer multi-objective programming problem using different values of \(\epsilon \). LINGO software is applied to find the optimal solution of single objective non-linear mixed integer programming problem. Finally, a numerical example is given to illustrate the proposed programming problem.KeywordsMulti-choice programming problemMulti-objective programming problemProbabilistic programming problemFractional programming problem\(\epsilon \)-constraint method

Converting general nonlinear programming problems into separable programming problems with feedforward neural networks

Constrained optimization for microbial fermentation was studied. For optimization, we used not the maximum principle but a nonlinear programming method because of the need to consider many metabolic reactions. In the case of L-lysine fermentation, the optimization problem in L-lysine production was formulated as a nonlinear programming problem. In general, the state equations based on material balances are represented as differential equations, but such equations which are dependent on time can not be applied to a nonlinear programming problem. Therefore, the state equations were made discrete in a time base, and a new single vector which is not dependent on time was substituted. From these formulae, the objective function and the constraints using nonlinear programming problem were defined as the amount of L-lysine produced, and as a metabolic reaction model and empirical equations, respectively. Computer program was developed to solve this constrained nonlinear programming problem. The applied algorithm of the computer programming was a sequential quadratic programming method (SQP method). When the constrained nonlinear programming problem is solved using the SQP method, the maximum amount of L-lysine produced and the optimal feeding rate of L-threonine could be calculated. From the calculated results, it was clear that introduction of the equality and inequality constraints was easy. L-Lysine at a concentration up to 75.3 g/l could be produced when the fermentation was carried out under optimal conditions.