Sure it is possible (in theory).

Moon's albedo is 12%

Photovoltaic panel efficiency is around 19% (commercially available) and can go up to 40% with more exotic technologies.

Assuming Moon gets the same amount of solar radiation as Earth, surface receives 1367 watts per square meter, 42% of which is visible light, which gives us 574 watts per square meter to play with. LEDs should beam back 69 watts. Assuming that we are using commercially available LEDs with 50% efficiency, 1 square meter should house 138 watts of LED power. This is a lot, but our bulbs will cover only a fraction of surface. The rest can be used for solar panels. Solar panels, on the other hand, will give us 229 watts per square meter.

During lunar day, panels will be baking in sunlight, converting it to electricity, which would be stored in batteries (do we have enough lithium on Earth? Hmm...) during the night, the bulbs will turn on, creating illuminated Moon face.

Also note that while solar panels can cover 100% of Moon surface, LEDs need to be installed only on the visible side, which should double the energy balance in our favor.

P.S. Calculations above assume that lunar LEDs work just like the Moon's surface, i.e. their emission is omnidirectional. Our efficiency can be improved A LOT is we are allowed to beam light only in Earth's direction.

I wouldn't touch the actual moon, it's 384,400km away, it seems much cheaper & within reach to make your own "equivalent" using an array of satellites (maybe only 320km - 600km away) with very bright lights, maybe in a geosynchronous orbit (approximately 35,786 km away) to keep them at least visible every night, using each one like a single pixel in a very large "display screen".

Either individual satellites just close enough together to look right, or tethered together with cables or filaments, or an extremely large single array or framework of bright lights.

So you end up with a virtual LED display in space. Perfect for displaying messages, or depending on the density of lights, even any picture or video.

With unlimited resources & energy, powering them should be within reach today with either solar power & batteries, or nuclear, or maybe even a Tesla-esque wireless power transmission from earth. Taking the "unlimited resources" more literally, then the pixel-satellites could even be brighter than the actual moon, so a "sky television" could even be seen during the day.

Some satellites are already visible from the earth now (and I'm pretty sure they don't even have any purpose-built lights aimed at the earth). Here's an image of some from How to See and Photograph Geosynchronous Satellites:

Just imagine a few million of them, tied together in a giant "screen" array, with unlimited energy for bright colours, and you've got your emperor's face, and propaganda, and a moving zooming or even exploding image of the moon, or Mars, or Jupiter, or anything really.

Here's a hopefully poor example using 160 computer keyboards (each with maybe 100 led lights), but it should give an idea of what's possible with even just 160,000 lights (from here, video here or directly on YouTube):

Numbers, numbers

• 
  

  The actual Moon surface is quite dark; the albedo of the Moon is 0.136. This means that the Moon reflects only 13.5% of the sunlight it receives. Moreover, the reflection is diffuse, that is, the reflected light goes all over the place, not only towards Earth.

  
  
  

  In order to move light from the far side of the Moon to the near side in the form of electric power, we need to (1) capture the energy of the light and convert it to electric energy, (2) transfer the electric energy to the near side, and finally (3) convert the electric energy into light. The overall efficiency of the process must at least match the 13.6% achieved by the Moon rocks through reflection.

  
  
  

  Can this be done?

  
  



• The efficiency of a decent photovoltaic panel is about 20%, meaning that the panel converts 20% of the incoming light energy into electric energy.




• The efficiency of a white LED lamp is currently around 15%, but 20% efficient lamps exist; the theoretical maximum luminous efficacy of a white LED is about 40%. Let's say that the emperor's scientists have achieved the capacity to make white LEDs with 30% luminous efficacy.




• Let's put the efficiency of electric power transmission at 90%.



• 
  

  Overall this gives 20% (light-to-electricity conversion) × 90% (transmission) × 30% (electricity-to-light conversion) = 5.4% overall efficiency. This means that, at best, the artificially illuminated new moon will have about 40% of the luminosity of the full moon; in photographic terms, that's a difference of about 3.5 stops of exposure; in astronomical terms, this is a difference of one magnitude.

  
  
  

  How visible is the difference in luminosity? Here is an image showing a normally exposed full Moon and a copy with the luminosity reduced to 40%.

  
  

The photograph of the Moon on the left is exposed so that the highlights are close to the maximum value, without exceeding it. The Moon on the right is the same image, digitally manipulated to make the Moon have 40% of the luminosity. Own work.

• But what about the phases of the Moon? Won't there be a marked difference in luminosity between the naturally lit and the artificiall lit parts? Yes, there will be a one-magnitude, or 3.5 stops, difference; visible, but hey, it very much better than the current situation.




• But what about the non-uniform illumination of the photovoltaic panels? True, the Moon is spherical, and the conversion efficiency of the photovoltaic panels on the far side will vary between the theoretical maximum when the Sun is up in the sky to zero when it is on the horizon; this will bring the available power down a factor of two, and make the artificially illuminated part even darker. Actual calculation remain as an exercise for the reader; however, overall we can confidently say that we can build a decent artificial lunar illumination system for our glorious and much beloved emperor.

Much cheaper option:

Every evening, send a large LED panel into the upper troposphere, e.g., with balloons and haul it down every morning. Since the panel is a lot closer to the ground, it can affordably be a lot smaller in cross section than the moon.

Plus side:

You can then rig the LED panel like a normal TV or monitor.

Minus side:

• It will always remain in the same place.


• There will normally be two "moons" in the sky.

EDIT: It can be done, see my second edit below.

Fundamentally can't be done. This issue was parodied at XKCD. The gist of the problem is this: you can't duplicate the firepower of the sun, especially if you're using a solar-based power system that isn't 100% efficient. Even if it was. It would need to acquire 100% of the solar energy that would hit the moon during a full moon, transfer that power perfectly (100% efficiency... the engineer within is starting to weep) to LEDs, which can emit the collected power as photons with 100% perfect conversion (oh, the pain!).

Can't be done without serious power. Serious. Check out the link. Serious.

Edit:

Also, remember that where there's solar panel, there isn't LEDs. You can hide the batteries underground and put the panels on the backside of the moon (it's tidally locked), but that means you must capture and store enough power to illuminate all those buka-watt LEDs for each night. Serious.

Edit:

OK, Shadoweze has piqued my curiosity. Lunar albedo for a full moon is 0.12. Albedo is the ratio of energy received to energy reflected. The sun bathes the moon in 1kw/m. So the reflection, what we need to achieve, is 0.12kw/m.

The full-moon lunar surface is 10¹³ m². That means we need to generate 1.2E12 watts or 1.2 terrawatts. The most efficient solar panel in 2018 has a 22.2% efficiency. That means for every kw of solar energy we'll actually have only .222 kw to work with. That's twice-ish what we need, so far so good.

Average lunar light is about 0.015 foot-candles or about 0.0019 lumens per m² for 0.016 lumens-per-watt of lunar emittance. 2018's most efficient LED is 105 lumens-per-watt. Good news! We don't need to cover every inch of the earth-side of the lunar surface!

Better news is that for each pass of the moon in front of the sun, there isn't a commensurate "full-moon" pass for the earth. The moon varies from a new moon (100% LED use) to a full moon (0% LED use). I'm absolutely wrong with the assertion I'm about to make, but to keep this from becoming a full dissertation, let's assume we only need to store 50% of the power needed to hold a full moon all the time and there's enough space between the LEDs that we can use detectors to shut off the LEDs we don't need during each phase of the moon. (And I'm ignoring the fact that we only need to turn the LEDs on when the emporer is in the night cycle. Who cares what the peons see, right?)

OK, I'm convinced. Shadowzee's right. It can be done. It might need enough battery mass to shift the moon's orbit... but it can be done.

Why is the XKCD no longer relevant? It's emitted light from the earth reflecting off the moon. That takes a ton more power, and I'd ignored it.

Are there any insurmountable technical problems?

Not really, unless you are under some time constraint. Moon reflectivity is around 0.12, and considering directed reflectivity that goes down to around 0.02. In other words, less than 2% of the incident solar light gets reflected towards the Earth.

So it's possible to cover the Moon with laser LED arrays and solar panels; even with very low efficiency and transmission losses, the Sun will supply energy enough to power a virtual moonlight beam.

Of course most of the material will need to be mined in situ. This means replacing copper with aluminum and magnesium wherever possible. On the other hand, vacuum refinement of silica would be easier than on Earth. Solar panel fabrication would be an ongoing project, because the solar wind might age panels faster than normal.

The Moon has a visual radius as seen from Earth of around 16 arc-minutes, so it covers 800 arc-minutes squared. The best resolution the human eye is capable of is around .7 arc-minutes; make that 0.5 and you have a radius of 32 pixels, requiring around 3220 beam emitters. The "image" thus obtained will be indistinguishable from the Moon by the naked eye. You do not need to cover the whole Moon with LED panels (or solar panels).

The total visible light from the Moon towards the Earth has been plausibly calculated by the Internetz at 1/436000 of the Sun, the latter being 550 W/m^2 in the visible range.

So we can assume the total output required of each beam emitter to be around 50 MW. Solar panels can supply around 200 W/m^2, requiring a minimum of 25-50 hectares of solar panels for each tower, or 160,000 hectares in all. We may actually need up to about three times that to cover the new moon stage (when the whole 3220 emitters are powered by a ring of solar farms just beyond the terminator, with equivalent power transmission line lengths of 2800 km in length and losses thereof approaching 20% for aluminum-magnesium lines).

As already stated - lighting up the surface of the Moon with LED's to mimic a full one can't be done with just solar panels and LED's.

"Possible" other options?

• Nuclear power to provide the electricity - @Gary Walker pointed out the amount of Thorium on the moon to use for fuel. Benefit of not being limited by incoming sunlight


• Earth based generators beaming power to the moon via microwaves?


• Network of giant mirror satellites to reflect sunlight onto the moon when it's behind earth


• During non-full moons, fly a massive plane/drone equipped with a giant LED screen between the Emperor's location and the moon, to mimic the appearance of a full moon. Added bonus of being able to display messages

Aiming makes a huge difference with LEDs

This adds to several answers which do the heavy calculations to duplicate the moon's albedo with LEDs. But the LEDs don't need to work nearly that hard.

The only light that matters is the light that's aimed at the Earth.

Since LEDs only emit about a 140-160 degree wedge of light, they lend themselves to aiming with lenses, which are extremely efficient. You narrow its radiant angle to just cover the earth. This dramatically reduces the energy required, by a factor of very roughly 99%, and that makes this a whole lot more practicable.

Since the moon is in synchronous rotation, you will only need to aim the LED once, you don't need heliostats to track the Earth.

Recently a Chinese company had proposed plans to put fake moon into orbit.

This is a direct Quote from BBC

According to the People's Daily state newspaper, officials at a
private aerospace institute in Chengdu want to launch this
"illumination satellite" in orbit by 2020, and say it will be bright
enough to replace street lights.

Why i pointed to this concept/idea?

Because in this scenario you do not need solar panels and leds which i think will cut costs and "EMPEROR" highness would be pretty much happy. No major capital burning here!

When and How ?

The satellite would be put to orbit by 2022 or so and would be known as the "Illumination Satellite", which would have reflective panels to reflect sunlight similar to putting a gigantic mirror in space.
It would be in a geostationary orbit roughly 37,000km from Earth.

This is not the First of its kind mission (Quote from same BBC article)

In 1993, Russian scientists released a 20m-wide reflector from a
supply ship heading to the Mir Space Station, which was orbiting at
between 200km and 420km.

Znamya 2 briefly beamed a spot of light about 5km in diameter to
Earth. The light marched across Europe at 8km/hr, before the satellite
burned up on re-entry.

Side-effects / Disadvantages :

• May affect natural sleep cycle in Humans.


• Nocturnal creatures would be
  affected drastically.

This is impossible for a multitude of reasons.

First of all, the moon doesn't really receive enough sunlight to power LEDs covering half of its surface. The amount of energy a single square inch of modern solar panel produces in an hour is only about 0.1 watts, whereas the amount of energy required to power a single modern LED for an hour is 6 watts. That means that an 8"x8" solar panel would be required to constantly power an LED that takes up a fraction of a square inch of space. Area-wise, the bright side of the moon just isn't big enough to power its opposite half.

Second, there is the issue of transporting the power you get to the LEDs. Every cable that we have loses power for every foot that it travels, which is a natural limit on how far power can be transported. Even with fiber optic -- the most efficient path we could use with 2018 technology -- the power loss experienced over the hundreds of miles of the moon's diameter would be too great for anything to reach the other end.

I can't put this in the original question because it might invalidate some answers. Therefore I'll put it as an 'answer' to my own question.

White Solar Panels

It’s being touted as a ‘revolution in renewable energy architecture’ –
the world’s first white solar panel with no visible cells or
connections.

https://www.energymatters.com.au/renewable-news/white-solar-panels-em4579/

Using these we could cover the side of the moon facing us and immediately get a much brighter moon. This would also make the unlit side somewhat visible. The LEDs could be inserted through holes on the panels and any stray illumination from them would also be reflected.

This is incomplete but it would supplement answers by those who have assumed solar panels must be black.

Note: this answer was thought of before noticing the stipulation of "needs 2018 technology". I'm still including it because it would be exactly the kind of thing an worldwide emperor who wants a permanent full moon would approve of.

Move the moon to the L2 Sun-Earth Lagrange point

the L2 Sun-Earth Lagrange point is one of the 5 points in an orbital pattern where you can place objects in a relatively stable location relative to the 2 larger objects in the pattern. It is located about 1 million miles away from Earth in the opposite direction from the Sun. Moving the moon there means it should always be full moon.

Moving a gigantic natural satellite like this is impossible with our current tech. It would also have an absolutely massive impact on Earth's ecosystem: tides would change, no more lunar cycle for nocturnal animals, it would break the Earth-Moon Lagrange points and it would forever remove the possibility of solar eclipses.

Ring Roads Around The Moon

There have been several Answers regarding photovoltaics and LEDs, which have missed a crucial point : no single device both creates and ingests light. If you have static devices on the lunar surface, you would need to alternate them on the Lunar ground, generating a fraction of the possible light. Directionality might improve that fraction.

Another possibility is to have moving devices. Imagine a large number of rings around Luna, built of railroads. Each ring is covered in rail cars that either receive light (photovoltaics) or creates light (LEDs). Electricity is transmitted through the rail, suitably insulated or through super-conductivity. Luna's synodic period is about 27.3 days, and Luna's circumference is 6786km, so the fastest rail at the equator needs to travel at 248km per day, or about 11km an hour.

Never stopping moving, with different LED cars set to different colors / brightnesses to emulate the ground underneath them.