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$$\frac{-\Delta G}{T}=\Delta S_{universe}$$
(This equation applies under isobaric and isothermal conditions.)
I understand that if $\Delta G$ is positive, the reaction is nonspontaneous, and adding that amount of energy to a system can drive that reaction to occur. However, if you’re increasing the Gibbs free energy of the system, then by that equation, the entropy of the universe would be decreasing, which would violate the Second Law of Thermodynamics. Can anyone explain this?
When you add energy from somewhere, let's say a battery, you are creating a greater amount of entropy there (in the battery or whatever). That's all there is to it. The Second Law is not violated.
The electrolysis process is not non-spontaneous. Yes, it would not occur in the absence of the battery. But the fact that it does occur with the battery there must tell you that overall it is spontaneous. The fact that you are coupling a strongly spontaneous reaction (discharge of the batter) with a non-spontaneous reaction (the electrolysis) means that overall, the electrolytic process is spontaneous.
It's incorrect to treat the electrolysis as if it occurs without the battery.
Any reaction depends on two factors: enthalpy and entropy, and the summation of which is the change of the Gibbs energy by equation (for isobaric and isothermal process):
$$\Delta G = \Delta H - T \Delta S \qquad(1)$$ or, after manipulations we have
$$\frac{\Delta H-\Delta G}{T} = \Delta S \qquad(2)$$
The difference between equations (2) and one you've represented is the last one describes process of the hypothetical system (or real -> Universe) which always become to the initial state and there's no change of enthalpy: $$\Delta H = 0 \qquad(3)$$
If you will try to change Gibbs energy: $$\Delta G = G_2 - G_1 \qquad (4)$$ where$$G_i = H_i - TS_i \qquad (5)$$ or $$ G_i = U_i + PV_i - TS_i\qquad (6)$$ you have to change internal energy or volume of the system which will effect on enthalpy and the change of it: $$\Delta H \neq 0 \qquad(7)$$ and you will get the equation (2)
Answering on your question: handle the change of enthalpy (7) of the reaction.
A non-spontaneous reaction does not proceed without the continuous supply of energy: A spontaneous reaction is one that, once started, continues without assistance. Under a particular set of conditions (e.g. room temperature), the reverse of a spontaneous reaction will always be non-spontaneous. For example, if A to B is spontaneous at 300 K, B to A is non-spontaneous at 300 K. However, a substantial increase in temperature (achieved by adding energy) often reverses spontaneity, so that (in our example) B to A is now spontaneous at (say) 1000 K and A to B non-spontaneous at 1000 K.
Electrolysis is an unusual tactic for forcing some non-spontaneous reactions to occur at room temperature. For example, the decomposition of water at room temperature is non-spontaneous, but a continuous supply of electricity causes continuous production of hydrogen and oxygen gases. Part of the electricity provides the energy to break the O-H bonds in water and part of the electricity is used to warm the water reservoir and the laboratory; the latter is essential for the entropy of the universe to increase without which the non-spontaneous reaction could not occur.
