Expert habitat: a colonization conjecture for exoplanetary habitability via penalized multi-objective optimization-based candidate validation

Abstract

The rate at which interstellar habitable planets are being discovered would naturally warrant consideration and exploration of a number of related issues. While the physical conditions that can support persistent contact demand structural similarity of an extra-solar planet (exoplanet) to Earth, and the necessary bio-chemical conditions needed to sustain life, potential for interstellar trade and extraction remain valid nonetheless. Based on the aspects that are commonly referred to as Earth similarity and habitability, we propose a novel bi-objective optimization framework as a tool to measure Earth similarity score (CDHS). This is followed by conjectures on possible interactions between Earth similarity and habitability, via two variants of penalized multi-objective particle swarm optimization, namely speed constrained multi-objective PSO (SMPSO) and a novel variant of multi-objective quantum PSO (MOQPSO). The optimization framework dispenses of classical gradient descent/ascent approach (GD/GA) by replacing it with SMPSO and MOQPSO. The approach to the input–output relations commonly adopted in production economics can be a natural influence for modeling habitability in exoplanets. An insightful demonstration establishes this claim. The scores reveal potentially habitable planets for interstellar trade. An analytical model of colonization in an exoplanet is also presented where we derive conditions for interstellar resource extraction and the volume of trade as function of time.

Appendices

Appendix

Convergence for hypervolume terminated QPSO (HT-MOQPSO)

We assume the quantum delta potential model of PSO where,

$$\begin{aligned} |\psi |^2 \mathrm{d}x\mathrm{d}y\mathrm{d}x = Q\mathrm{d}x\mathrm{d}y\mathrm{d}z. \end{aligned}$$

(25)

$|\psi |^2$ is the probability density function satisfying

$$\begin{aligned} \int _{-\infty }^{+\infty } |\psi |^2 \mathrm{d}x\mathrm{d}y\mathrm{d}x = \int _{-\infty }^{+\infty } Q \mathrm{d}x\mathrm{d}y\mathrm{d}x =1. \end{aligned}$$

(26)

The state function, $\psi (x,t)$ is described by Schrödinger equation. Consider H as the Hamiltonian operator, for a single particle of mass m in a potential field V(x), given as

$$\begin{aligned} H = -\frac{h^2}{2m}\varDelta ^2 + V(x), \end{aligned}$$

(27)

where h is the Planck’s constant. Let us consider the time dependent and time independent Schrodinger equation to arrive at the variant describing the delta potential well of QPSO: $H\psi = ih\frac{\partial \psi }{\partial t} ; \text { time dependent variant}$ and $H\psi = E\psi ;$ time independent variant where E $=$ Energy eigen value and V(x) $=$ $-\gamma \partial (x - p) = -\gamma \partial (y)$; y $=$ $x -p$. p is the center of the attraction potential field. This is essential for stability and bound state of the potential. Using the above variants, we can write

$$\begin{aligned}&\left[ -\frac{h^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}y^2} - \gamma \partial (y)\right] \psi = E\psi \\&\quad \implies \frac{\mathrm{d}^2\psi }{\mathrm{d}y^2} + \frac{2m}{h^2}\gamma \partial (y)\psi = -\frac{2m}{h^2}E\psi \\&\quad \implies \frac{\mathrm{d}^2\psi }{\mathrm{d}y^2} + \frac{2m}{h^2}[\gamma \partial (y) + E]\psi = 0\\&\quad \implies \frac{\mathrm{d}^2\psi }{\mathrm{d}y^2} - \beta ^2\psi = 0 ; \text {where } \beta = \sqrt{\frac{-2mE}{h}}. \end{aligned}$$

Let L $=$ $\frac{1}{\beta }$. The wave function (normalized) and the probability density function can be represented as $\psi (y) = \frac{1}{\sqrt{L}}e^{-|y|/L}$ and $Q(y) = \frac{1}{L} e^{-2|y|/L}$, respectively. The termination condition on the MOQPSO algorithm based on hypervolume is based on the assumption that successive iterative computation of the area including solution set (Pareto front) would converge, i.e., the differences between the successive areas bounded by $\epsilon$ implying the swarm movement being restricted in an $\epsilon$ - neighborhood to guarantee convergence. Since, the probability density function is computed already. We arrive at the expression stating the difference in hypervolume. $||A_{i+1} - A_{i}|| = \int _{-\epsilon }^{+\epsilon } ||Q(A_{i+1}) - Q(A_{i})|| \mathrm{d}y \rightarrow \epsilon$. As $\epsilon \rightarrow 0$ when $i \rightarrow \infty$, HT-MOQPSO is guaranteed to converge asymptotically.

Modeling constraints using penalties

We represent all strict inequality and equality constraints as non-strict equality constraint as described by Ray and Liew [49]. We convert strict inequality constraint of the type $g'(x)\,<\,0$ to a non-strict inequality constraint g(x) by introducing an error term $\epsilon$ such that $g(x) = g'(x) + \epsilon \,\le 0$. By introducing a tolerance value $\tau$, we convert equality constraint of the form $h(x) = 0$ to $g(x) = |h(x)| - \tau \le 0$. For a solution $p_i$, let $c_i$ denote the vector of constraint values. Then $c_{ik} = max(g_k(p_i),0)\,\forall \,k=1,2,3,\ldots ,m$. When $c_{ik} = 0$, then solution $p_i$ lies in the feasible region of the search space.

Applying this rule, constraints under CRS can be translated to

$$\begin{aligned}&-\phi + \epsilon \le 0,\,\, \phi -1 + \epsilon \le 0\,\, \forall \,\phi \in \{\alpha ,\beta ,\delta ,\gamma \}, \end{aligned}$$

(28)

$$\begin{aligned}&|\alpha + \beta -1| - \tau \le 0, \,\, |\delta + \gamma -1| - \tau \le 0. \end{aligned}$$

(29)

Under DRS, we replace (29) with $\alpha + \beta + \epsilon -1 \le 0,\,\,\delta + \gamma + \epsilon -1 \le 0$. We impose these constraints through the use of penalty methods. In penalty methods, we augment the objective functions with penalty functions that “penalizes” a candidate solution when it violates any of the constraints. In case of a minimization problem, penalty functions return a large positive value, when a candidate solution moves outside of the feasible region, that gets added to the base objective function. This, in turn, makes the objective function large and undesirable and hence, making the candidate solution weak.

We define the following penalty functions:

$$\begin{aligned}&\psi (x) = {\left\{ \begin{array}{ll} 0,&{}\quad \text {if } |x| - \tau \le 0,\\ k_1.|x|,&{}\quad otherwise \end{array}\right. },\\&\varOmega (x) = {\left\{ \begin{array}{ll} 0,&{}\quad \text {if } x + \epsilon \le 0,\\ k_2.|x|,&{}\quad otherwise. \end{array}\right. } \end{aligned}$$

$k_1$ and $k_2$ are penalty factors. Larger the penalty factors, the more severe the penalty is. Using functions $\psi$ and $\varOmega$, we augment objective functions (4) and (5) under CRS condition as

$$\begin{aligned} PY_i= & {} -Y_i + \psi (\alpha + \beta -1) + \varOmega (-\alpha ) + \varOmega (\alpha -1)\nonumber \\&+ \varOmega (-\beta ) + \varOmega (\beta - 1) \end{aligned}$$

(30)

$$\begin{aligned} PY_s= & {} -Y_s + \psi (\delta + \gamma -1) + \varOmega (-\delta ) + \varOmega (\delta -1)\nonumber \\&+ \varOmega (-\gamma ) + \varOmega (\gamma - 1). \end{aligned}$$

(31)

Using these augmented objective functions, the constrained optimization task (8) subject to (9) is equivalent to the unconstrained optimization task: $\min _{\mathbf {x}} \mathbf {f}(\mathbf {x})=[PY_i,PY_s]$. In our experiments, we set k1 and k2 to $10^{12}$ and make $\epsilon$ and $\tau$ equal to $10^{-8}$.

Hyper-parameter tuning

Tuning and improvising parameters such as max and min velocity, learning factors such as cognitive and social factors, inertia weight etc. is not possible through the methods that the classes in Jmetalpy provides. Tuning of these parameters is really crucial for the algorithm to converge. With respect to the range of the functions that we are trying to optimize and the constraints that are imposed on the search space, the algorithm, with the default parameters provided by the library, did not yield desirable solutions. Most of the solutions in the solution set were outside of the feasible region of the search space. We suspected that $V\mathrm{max}$ was too large and $V\mathrm{min}$ was too small. And hence, some changes were made in $Jmetalpy's$ source code to make parameter tuning possible. Initially, for jth decision variable,

$$\begin{aligned} V\mathrm{max}_j= & {} \frac{\mathrm{upperbound}_j - \mathrm{lowerbound}_j}{2.0}; \\ V\mathrm{min}_j= & {} -V\mathrm{max}_j, \end{aligned}$$

where $\mathrm{upperbound}_j$ is the largest allowable value for the jth decision variable and $\mathrm{lowerbound}_j$ is the smallest allowable value for the jth decision variable. For our problem, these values were changed to

$$\begin{aligned} V\mathrm{max}_j = \frac{\mathrm{upperbound}_j - \mathrm{lowerbound}_j}{1000.0} \end{aligned}$$

with $V\mathrm{min}_j$ still set to $-V\mathrm{max}_j$. Learning factors w, $C_1$ and $C_2$ are sampled from a uniform distribution with specified ranges, i.e., $w \sim U(w_{\min }, w_{\max })$, $C_1 \sim U(C_{1\min },C_{1\max })$, and $C_2 \sim U(C_{2\min }, C_{2\max })$. We set $w_{\min } = w_{\max } = 0.1$, $C_{1\min } = 0.1$, $C_{1\max } = 0.5$, $C_{2\min } = 0.8$ and $C_{2\max } = 1.5$. The swarm size is set to 100.