We present a rigorous and comprehensive survey on extensions to the multicriteria setting of three well-known scalar optimization algorithms. Multiobjective versions of the steepest descent, the projected gradient and the Newton methods are analyzed in detail. At each iteration, the search directions of these methods are computed by solving real-valued optimization problems and, in order to guarantee an adequate objective value decrease, Armijo-like rules are implemented by means of a backtracking procedure. Under standard assumptions, convergence to Pareto (weak Pareto) optima is established. For the Newton method, superlinear convergence is proved and, assuming Lipschitz continuity of the objectives second derivatives, it is shown that the rate is quadratic.

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In 2004, Gran˜a Drummond and Iusem proposed an extension of the projected gradient method for constrained vector optimization problems. Using this method, an Armijo-like rule, implemented with a backtracking procedure, was used in order to determine the step lengths. The authors just showed stationarity of all cluster points and, for another version of the algorithm (with exogenous step lengths), under some additional assumptions, they proved convergence to weakly efficient solutions. In this work, first we correct a slight mistake in the proof of a certain continuity result in that 2004 article, and then we extend its convergence analysis. Indeed, under some reasonable hypotheses, for convex objective functions with respect to the ordering cone, we establish full convergence to optimal points of any sequence produced by the projected gradient method with an Armijo-like rule, no matter how poor the initial guesses may be.

The LoRa physical layer is one of the most promising Low Power Wide-Area Network (LPWAN) technologies for future Internet of Things (IoT) applications. It provides a flexible adaptation of coverage and data rate by allocating different Spreading Factors (SFs) and transmit powers to end-devices. We focus on improving throughput fairness while reducing energy consumption. Whereas most existing methods assume perfect SF orthogonality and ignore the harmful effects of inter-SF interferences, we formulate a joint SF and power allocation problem to maximize the minimum uplink throughput of end-devices, subject to co-SF and inter-SF interferences, and power constraints. This results into a mixed-integer non-linear optimization, which, for tractability, is split into two sub-problems: firstly, the SF assignment for fixed transmit powers, and secondly, the power allocation given the previously obtained assignment solution. For the first sub-problem, we propose a low-complexity many-to-one matching algorithm between SFs and end-devices. For the second one, given its intractability, we transform it using two types of constraints' approximation: a linearized and a quadratic version. Our performance evaluation demonstrates that the proposed joint SF allocation and power optimization enables to drastically enhance various performance objectives such as throughput, fairness and power consumption, and that it outperforms baseline schemes. 1 DRAFT 2 I. INTRODUCTIONA wide range of applications will be enabled by the advent of Internet of Things (IoT) technology, among which smart cities, intelligent transportation systems and environmental monitoring.Given the expected proliferation of such IoT devices in the near future, providing tailored wireless communication protocols with high spectral efficiency and low power consumption is becoming more and more urgent. Indeed, many of these services will depend on the future IoT Wireless Sensor Networks (WSNs), supported by the newly developed Low-Power Wide-Area Network (LPWAN) technologies such as LoRa, SigFox or Ingenu [2-5]. The LoRa physical layer uses the Chirp Spread Spectrum (CSS) modulation technique, where each chirp encodes 2 m values, for Spreading Factor (SF) m = 7 to 12 [6], and which allows multiple end-devices to use the same channel simultaneously. Based on the LoRa physical layer, LoRaWAN defines the MAC layer protocol standardized by LoRa Alliance [7]. It is an increasingly used LPWAN technology, as it operates in the ISM unlicensed bands and enables a flexible adaptation of transmission rates and coverages under low energy consumption [6]. The LoRaWAN architecture is a star topology,where end-devices communicate with the network server through gateways over several channels based on ALOHA mechanism, with duty cycle limitations [4]. In LoRaWAN, smaller SFs provide higher data rates but reduced ranges, while larger SFs allow longer ranges but lower rates [5].The main issue of LoRa-based networks such as LoRaWAN is the throughput limitation: the physical bitrate varies...